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Ind. Eng. Chem. Res. 1999, 38, 1432-1443
Dominant Variables for Partial Control. 1. A Thermodynamic Method for Their Identification Bjo1 rn D. Tyre´ us† DuPont Central Research & Development, Experimental Station, P.O. Box 80101, Wilmington, Delaware 19880-0101
Partial control is a decentralized control strategy whereby a number of economic operating objectives are controlled either at their setpoints or within a specified range by feedback control of a few dominant variables. In many processes that have been studied the economic variables are related to the flow and production rates in the process. These rates also determine the magnitude of the energy exchange between the plant’s different energy carriers. Thus, a thermodynamic process description can be used to identify the dominant variables affecting the rate of energy exchange within the plant. This approach to dominant variable identification is very efficient in that it focuses on a single expression containing all the dominant variables for each process unit and it ignores all irrelevant process states. In this first part the method is demonstrated on simple units as well as on the more complicated fluidized catalytic cracker. In part 2, the method is applied to the Tennessee Eastman challenge process. Introduction Partial control is a form of decentralized control. In this study, decentralized control refers to strategies in which the controlled variables are explicitly paired with the manipulated variables such that the main feedback loops are easily identified. Furthermore, each loop in a decentralized control strategy has its separate feedback controller. This should be contrasted to a centralized strategy which typically uses a single, model-based controller common to multiple loops in the process. In most complex processes the number of economic operating objectives far exceeds the degrees of freedom for control. This implies that the objectives cannot be all simultaneously controlled to their setpoints. In centralized control strategies this problem is addressed by optimizing an objective function that represents all the control objectives. The alternative approach is to use partial control whereby a number of the economic operating objectives are controlled either at their setpoints or within a specified range by feedback control of a few dominant variables. A typical example of partial control is the feedback regulation of temperature in an exothermic reactor. This single control loop simultaneously affects reactor stability, productivity, selectivity, and yield. Similarly, the control of one sensitive tray temperature in a conventional distillation column affects the purity of both products as well as the unit’s operating cost. Partial control strategies are common in industry because they are straightforward to implement, robust, cost-effective, and easy to understand. Applications of partial control date back to the early use of automatic process control when technology limitations and cost factors made it necessary to find a few simple measurements and actuators to control the process. With modern sensors and distributed control systems (DCS) the technology and cost barriers have been lowered but partial control remains a favored strategy for reasons of simplicity and robustness. † E-mail:
[email protected]. Tel.: 302-6958287. Fax: 302-695-2645.
The formal description of partial control was first put forward by Shinnar1 and recently illustrated by Arbel et al.2,3 through extensive case studies on an industrialscale fluidized-bed catalytic cracker (FCC). Formal definitions of the terms used to describe partial control systems have been provided by Kothare et al.4 However, despite the widespread use of partial control and the early introduction of the ideas behind it, there have been few papers published on the subject. The reason is most likely due to the technique’s empirical and heuristic nature. For example, the control strategies for an FCC described by Arbel et al.2,3 had evolved from heuristic guidelines and long experience with the process. It is not unusual that an effective control strategy, formulated on the basis of experience (from simulation or process operation), can be identified in hindsight as a partial control strategy. However, it is not at all obvious how to design a partial control system a priori for a new process. For example, it is seldom clear how many dominant variables need to be controlled to meet the overall operating objectives. Indeed, it is not even obvious that a process will have a small set of dominant variables and how these can be identified. At the opposite end of the spectrum from partial control we find model-predictive control techniques and control synthesis methods based on thermodynamically extensive variables. These techniques have solid theoretical foundations as well as formal procedures to construct the control systems without requiring much insight or experience with the process. Extensive variable-based techniques have the added advantage of linking process control to process design. Their major shortcomings are complexity in implementation, generally higher costs, lack of transparency, and sometimes lack of robustness. It would thus be desirable to combine the virtues of partial control with the formal procedures of modelpredictive control or control based on thermodynamics. The goal is to be able to use simple dynamic models of the process to guide in the selection of dominant variables and sufficient partial control strategies. The purpose of this paper is to show how thermodynamics
10.1021/ie980619y CCC: $18.00 © 1999 American Chemical Society Published on Web 03/04/1999
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can help in identifying the dominant variables of the system. The paper is organized as follows. First, a general thermodynamic framework for representing process systems is introduced followed by a brief review of current control systems design methods using the extensive variables of the process. Next, the important features of partial control are examined and a connection is made between the intensive variables in the process and the plant’s economic objectives. This allows us to identify a class of candidate dominant variables. Finally, the ideas are illustrated on a flash tank, a distillation column, a continuous stirred tank reactor (CSTR) with two reactions, and a fluidized catalytic cracker. System Descriptions The key to using thermodynamics for the identification of dominant variables is an appropriate system description. The goal is to avoid having to deal with the full set of state variables representing a dynamic system and instead focus on a few pertinent equations that contain all the dominant variables. We can make progress along these lines by adopting the unified approach to physics developed by Burkhardt5 and Fuchs.6 Their approach is quite different from the traditional teaching of classical physics and thermodynamics and therefore requires a bit of an investment in time and effort to understand and appreciate. For example, traditionally we are accustomed to working with various energy forms (e.g., potential, kinetic, electrical, chemical, heat, work, etc.), but there is no need for this differentiation in the unified approach. Instead, energy is treated as a quantity that is carried by other fundamental variables such as electric charge or momentum. Similarly, in classical thermodynamics the concept of entropy is introduced as a state function that is related to heat (an energy form). In the unified approach the roles are reversed; entropy is the fundamental variable and energy is seen as being carried by currents of entropy. Central to the unified approach to physics is the use of a generalized balance equation to describe all physical processes. For example, the same balance equation is used to describe the transient behavior of an electrical circuit, a hydraulic device, and a chemical reactor. To make this possible, Burkhardt5 adopted the idea introduced by Falk et al.7 and Schmid8 that certain thermodynamic quantities behave as if they were “substancelike”. The word substance-like is thus attributed to any physical quantity that behaves like an actual substance. For example, the total mass of material within a process is a substance-like quantity because it measures what we already know as a substance. Similarly, the amount (mol) of chemical components and electric charge are two more examples of substance-like quantities. While theses examples involve variables we naturally associate with substances, this does not have to be the case. Other substance-like quantities are energy, entropy, and momentum. It may seem strange to think of momentum and entropy as “substance-like”, but conceptually these quantities behave as if they were a physical material. In fact, Schmid8 attributes three important characteristics to any substance-like quantity. First, such a quantity has a density such that the total amount of the substance-like quantity within a given system is obtained by adding or integrating the local densities over the entire volume representing the system. Second,
a substance-like quantity can flow across space and can therefore enter or leave the boundaries (surfaces) surrounding a system. Finally, each substance-like quantity has a balance (or continuity) equation that accounts for its existence over time. At first glance it may appear unnecessary to introduce the “substance-like” characterization for quantities that are already labeled extensive in classical equilibrium thermodynamics. Recall that an extensive property, such as volume or mass, is one which is additive in the sense that the value of the property for the whole system is the sum of the values for all the different parts of the system. It is thus obvious that all substance-like quantities are extensive. However, not all extensive variables are substance-like. Take the dimensions (length, volume, etc.) of a system as an example; these variables are extensive but not substance-like. The generalized balance equation, applicable to all substance-like quantities in any system under all conditions, can be demonstrated by writing the equation that describes the dynamic behavior of the molar content of a single component in a chemical reactor. The amount of component is a substance-like quantity because it has a density (concentration) and it can flow in and out of the reactor by diffusion or convection. Furthermore, we can write a continuity equation for the accumulation of the component within the reactor.
dni ) -Ini + Πni dt
(1)
There are three main terms in this generalized balance equation written in the nomenclature used by Fuchs.6 (I have deliberately chosen to use his notation to make it easier for the interested reader to refer to his textbook where these ideas are developed in more detail.) First, on the left-hand side the time derivative accounts for the accumulation or depletion of the substance-like quantity over time. Second, the negative term on the right-hand side captures the net flow of material leaving the system, and finally, the last term models the production (or destruction) of the substance-like quantity within the system. In the case of a component balance with chemical reaction all three terms are present. This is also true for entropy balances since entropy is always produced in real processes. However, mass, energy, and momentum balances lack the production term because these substance-like quantities are conserved. While the balance equation is completely general and applies to substance-like quantities under all conceivable conditions, it cannot be solved until we have quantitative expressions for the flow and production terms. Such relations are called constitutive equations and depend on the properties of the system studied as well as the conditions surrounding it. A constitutive equation is not general in the same way as the balance equation. Instead, constitutive equations describe how substance-like quantities affect the state of a particular dynamic system and how the quantities flow in and out of the system, depending on the system’s state. For example, the ideal gas heat capacities and the ideal gas law are constitutive relations that apply only to systems containing ideal gases and relate the extensive variables (e.g., internal energy, mole number, and volume) to the intensive variables (e.g., temperature, pressure, and molar concentration). Similarly, transport equations
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such as Fourier’s law for heat conduction and Fick’s law for material diffusion use gradients of intensive variables to determine the fluxes of substance-like quantities. Production terms also use constitutive relations. Here, combinations of the intensive variables (e.g., temperature, pressure, and concentration) help determine the production rate of chemical components from empirical kinetic rate expressions. If we accept that all physical systems can be modeled through a combination of balance equations and constitutive equations, we next ask which substance-like quantities should be used to describe the dynamics of a particular physical system. This question is answered by Schmid8 and Fuchs.6 It turns out that mechanical systems use momentum (linear and rotational) as their fundamental substance-like quantity, electrical systems use electric charge, chemical systems use the amount of component, and thermal systems use entropy as the independent substance-like quantity. From this list it may seem surprising that energy, also being a substancelike quantity, is not used to describe a separate class of systems. The reason is that the balance of energy is not specific to any particular process but is the universal quantity that applies to all systems. Fuchs6 refers to this as the energy principle. Energy can thus be seen as the “glue” that unites the descriptions of different systems. From this perspective it is not necessary to consider different energy forms (e.g., kinetic, potential, heat, etc.) but rather view energy as a substance-like quantity that is always carried and associated with another substance-like quantity. (See Falk et al.7 and Fuchs6 for more details.) For example, the flow of electric charge (current) carries energy when it travels through an electric circuit. Similarly, a flow of chemical reactants into a reactor carries energy that is later released during the reaction, and a flow of entropy carries energy in the form we know as heat from classical thermodynamics. The amount of energy carried with the flow of a particular substance-like quantity is the product of the quantity’s flow rate and its intensity (potential). For example, the intensity associated with momentum is velocity while electric charge has voltage as its intensity. Similarly, the intensity associated with entropy is absolute temperature whereas the chemical potential represents the intensity of a chemical component. The intensities form a subset of the intensive variables used in classical equilibrium thermodynamics. The idea of substance-like quantities as energy carriers allows for an important interpretation of processes. Since energy cannot be created or destroyed, we require that every system, in the long run, must export as much energy as was carried into the system. However, in many chemical processes we are dealing with chemical, electrical, thermal, and mechanical systems all at once, meaning that energy carried into a system does not have to leave with the same carrier with which it entered. In fact, it is most useful to consider a process or a unit operation as an exchanger or transceiver of energy between different carriers (see Falk et al.7). A couple of examples will illustrate this idea. First, Figure 1 shows a schematic of a fuel cell. Here, a flow of chemical components carries an energy current into the fuel cell. Part of this energy is released when the chemical reactions advance inside the cell. The power release (energy per unit time) equals the rate of chemical transformation (reaction rate) multiplied by the change
Figure 1. Energy exchange between a chemical component current and a charge current in a fuel cell.
Figure 2. Energy exchange between a chemical component current and an entropy current in an isothermal reactor with an exothermic reaction.
in the chemical potentials as a result of the chemical transformation. The released power is absorbed by a flow of electric charge that receives an increase in its intensity (voltage). The exchanged energy is carried out of the fuel cell by the electric current. The fuel cell is thus an energy exchanger between a chemical component carrier and an electric current carrier. A second example of energy exchange is an isothermal exothermic reactor illustrated in Figure 2. Again, a current of energy is carried into the reactor by the flow of reactants. Some of the inflowing energy is released inside the reactor by lowering the chemical potentials of the reactants to those of the products at a rate equal to the reaction rate. The released energy is used to create entropy at the temperature of the reactor. The exchanged energy leaves the system with the entropy current traveling with the cooling water. The reactor thus serves as an exchanger of energy between a chemical component current and an entropy current. Furthermore, the energy conservation principle requires that the product of the entropy current and its intensity (absolute temperature) equals the power release due to the chemical reactions. A final example is a distillation column illustrated in Figure 3. Here, entropy, at an elevated temperature, carries energy into the system through the reboiler. The influx of entropy plus entropy created inside the column leaves the system through the condenser at a lower temperature. The net energy exchange per unit time (power release) can be computed from the change in temperature across the column multiplied by the flow rate of entropy into the reboiler. Part of the energy released by the decline in temperature is used to elevate the chemical potential of the components in the feed stream to those in the product streams. The remaining energy is used to create additional entropy to account
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1435
Figure 3. Energy exchange between an entropy current and a chemical component current in a conventional distillation column.
for mixing processes and pressure drop over the trays. In this unit operation the significant energy exchange takes place between an entropy current and a current of chemical components. To summarize, a system concept based on the generalized balance equation applied to one or more substancelike quantities is adopted to focus on the variables that determine the rate of energy exchange within a process. In this framework most chemical systems are modeled by N component balances and one entropy balance. In flow systems with greatly varying flow velocities, a linear momentum balance is also required. To solve the balance equations, we require several constitutive equations that depend primarily on the intensive variables of the process. Finally, at steady state, the energy transported in and out of a process by different carriers must equal the internal power release and satisfy the overall energy balance. The idea of viewing unit operations as energy exchangers is central to the proposed method for the determination of dominant variables for partial control. Specifically, we will observe that the economic objectives of a process are tied to the rates governed by the constitutive equations of the system. Since the concept of internal energy exchange is common to all processes, we focus on this rate of exchange to determine the variables that tend to dominate the behavior of the system. Extensive Variable Control Before the system concepts described above can be put to use in analyzing dominance and partial control, a brief mention will be made of other thermodynamic approaches to control system design. In fact, the concept of substance-like quantities is quite helpful in understanding the thermodynamically motivated methods proposed by Georgakis,9 Ydstie and Viswanath,10 and Ydstie and Coffey.11 These methods are natural extensions to the industrially common problem of controlling levels in process vessels. It is well-known that inventories can be effectively controlled with simple proportional-only controllers. A system described by a conserved substance-like quantity such as total mass has a mathematical model that is a pure integrator. Feedback regulation of this quantity with a proportionalonly controller gives a first-order system which is always stable and has a smooth, uniform response to external disturbances. When a process requires several independent substance-like quantities for its description, we can attempt to provide simple feedback loops around all the
Figure 4. Control of thermodynamically extensive properties in a binary flash tank.
independent quantities. This is the essence of extensive variable or inventory control systems. The favorable stability properties of proportional-only level controllers will not necessarily apply unless all regulated substance-like quantities are conserved. However, we already know that the two most common quantities in process systems, mole numbers and entropy, are not conserved. How do we then proceed? One trick is to use internal energy instead of entropy as one of the independent substance-like quantities in the control scheme. In addition, when there are no reactions taking place, we also have conservation of the mole numbers of each component. We can therefore close simple feedback loops around the internal energy of the system and each of the component inventories and expect stable and smooth performance. This technique has been demonstrated for several process units by Ydstie and co-workers.10,11 Figure 4 gives an example of the decentralized control of a two-component vaporizing flash unit as developed by Ydstie and Viswanath.10 In this application, flow, level, and temperature measurements are used to compute the total mass and energy in the vessel as well as the net external flow of these quantities. The total mass is balanced by adjusting the liquid flow leaving the vessel, and the total energy is balanced by adjusting the steam to the heating coils. It is important to point out that the sensors must necessarily measure intensive variables (temperature, level, pressure drop, etc.) because substance-like quantities are confined to the system whereas intensive variables act like fields and reach outside the system, including the sensors. The flow rates and stream temperatures are then used to reconstruct the net mass and energy flow to the flash so that the inventory of mass and energy can be balanced. The scheme in Figure 4 thus measures intensive variables to estimate the extensive variables and their flow rates. Partial Control While the logic involved in developing extensive variable control strategies is straightforward, the resulting control structure tends to be complicated and potentially difficult to implement, depending on how feasible it is to obtain reliable measurements for all the variables required to estimate the extensive variables and their flows. We need only look at Figure 4 showing
1436 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 Table 1. List of Economic Objectives for an FCC
Figure 5. Partial control of a fluidized catalytic cracker.
the control structure for a simple binary vaporizing flash unit to imagine what the extensive variable control structure might look like for a complex, multicomponent reactor system such as a fluidized-bed catalytic cracker (FCC). In sharp contrast to this control complexity stands the application of partial control to a modern FCC as described by Arbel et al.2,3 and schematically depicted in Figure 5. Here, two decentralized temperature control loops not only stabilize this dynamically complex unit but also provide the opportunity to meet a range of operating goals by adjustment of the controller setpoints. The question is how can we arrive at an efficient strategy like the one shown in Figure 5 for an arbitrary process without having years of operating experience or familiarity with the unit? Before we attempt to answer this question, a brief review of partial control will be given. For more details see Shinnar,1 Arbel et al.,2,3 and Kothare et al.4 Partial control hinges on the observation that many process systems seem to have a few independent variables that tend to dominate the dynamic behavior of the entire system. The temperature in an exothermic reactor and sensitive tray temperatures in distillation columns have already been mentioned as good examples. Once these dominant variables have been identified, they should be measured as frequently as possible, preferably continuously. Then, a matching number of independent, rapidly acting manipulated variables must also be identified. In an exothermic reactor the cooling rate is used as a manipulator whereas distillation columns frequently use steam rate to the reboiler as the manipulated variable. FCC applications use the air rate and catalyst recirculation rate as two independent manipulated variables. The dominant variables are paired with the manipulated variables in an appropriate fashion and the loops are closed with conventional proportional-integral (PI) or proportional-integralderivative (PID) controllers. An interesting feature of partial control is that the dominant variables in a system seldom have intrinsic economic value. For example, we do not assess the quality of a refined product in terms of a tray temperature and we do not measure productivity and yield with the reactor temperature. However, the dominant variables, by their nature, influence most other variables in the system including those that bring financial value to the operation. As mentioned before, reactor temperature often has a dominant effect on production rate, selectivity, and yield, all of which are important to cost and revenue results. A partial control strategy takes
product
constraints
octane conversion alkylation feed rate gasoline yield light cycle oil yield and properties C2 isobutane, propylene-to-propane ratio
CO, NOx, SO2 in flue gas wet gas rate air rate catalyst circulation rate flue gas temperature riser temperature regenerator temperature
advantage of this relationship by allowing the setpoints of the dominant feedback controllers to vary in order to meet the economic objectives. For example, when the overhead product from a distillation column contains excessive amounts of high boilers, we lower the setpoint of the tray temperature controller. This reduces the steam to the reboiler and eventually shifts the column’s composition profile in a favorable direction. Similarly, when a high production rate in the plant is needed, we raise the reactor temperature controller setpoint. Thermodynamics and Partial Control We now seek a physical justification for partial control with the intent of being able to identify candidate sets of dominant variables from the thermodynamic description of the process. To do this, we first examine the nature of the economic objectives that we try to meet with partial control strategies. Table 1 shows a list of specifications for an FCC as published by Arbel et al.2 and Table 2 gives the list of specifications for the Tennessee Eastman challenge process published by Downs and Vogel.12 These lists represent the type of economic objectives found for many other chemical processes as well. For example, we are nearly always interested in controlling the production rate of the plant and we often desire to maximize this rate by operating close to some physical constraint. The economic objectives also revolve around conversion, yield, and selectivity and depend critically on product composition and quality. These objectives are determined by the rates of formation and removal of various components in the system. Similarly, operational constraints are often stated in terms of limitations on flow rates or on intensive variables such as pressure or temperature. Composition constraints are also common to guard against hazardous conditions, corrosion, or release to the environment. It thus appears that the economic objectives of a process are tied mostly to the flow and production rates of substance-like quantities and to the intensive variables that help establish these rates. To use thermodynamic models to guide in the identification of dominant variables, we should thus focus on the constitutive equations (rate-related) rather than on the balance equation (inventory-related). However, it is not clear on what constitutive equation we should concentrate. This is where it is useful to invoke the energy principle since it is applicable to all processes. The energy principle states that, at steady state, the net energy flow around any process must be zero. However, energy flows in and out with different carriers. The internal rate of exchange of energy between the various carriers will thus be taken as the most relevant rate indicator of a process and the process variables affecting the energy exchange will be considered dominant. A couple of examples will illustrate the new ideas presented here. We start by considering a partial control
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1437 Table 2. List of Economic Objectives for the Eastman Process product
constraints
costs
rate specified to 14 076 pph with a purity of 50% G rate specified to 14 076 pph with a purity of 10% G rate specified to 11 111 pph with a purity of 90% G maximum achievable production rate at 50% G maximum achievable production rate at 10% G maximum achievable production rate at 90% G
none < reactor pressure < 2895 kPa 50% < reactor level < 100% none < reactor temperature < 150 °C 30% < separator level < 100% 30% < stripper level < 100% product flow rate variability
