# Fundamentals of Process Modeling for ISA Certification of Automation Professionals (CAP) Tips

Here are the new sections on process modeling fundamentals for the next edition of the Guide to the Automation Book of Knowledge that is the primary resource for the ISA Certification of Automation Professionals (CAP) Program. The following includes text that may not fit into the page allocation for Chapter 12 on process modeling.

12.1 Fundamentals

Process models can be broadly categorized as steady state and dynamic. Steady state models are largely used for process and equipment design and real time optimization (RTO) of continuous processes. Dynamic models are used for system acceptance testing (SAT), operator training systems (OTS), and process control improvement (PCI). A steady state or dynamic model can be experimental or first principle. Experimental models are identified from process testing.  First principle models are developed using the equations for charge, energy, material, and momentum balances and equilibrium relationships and driving forces.

Steady state models can be multivariate statistical, neural network, or first principle models. Multivariate statistical and neural network models are primarily used to detect abnormalities in process operation. First principle steady state models are widely used by process design engineers to develop process flow diagrams and to a much lesser extent by advanced control engineers for RTO.

For steady state first principle models the derivative of the ordinary differential equations (ODE) is set equal to zero and a search of process outputs is conducted to satisfy the resulting ODE. Volumes are treated as perfectly mixed and completely uniform. Profiles are created by partitioning the volumes into subsections. Partial differential equations provide a more detailed analysis of changes in process temperature and composition depending upon location within the volume and are principally used to study mixing patterns. First principle models iterate until a final solution is reached giving the final condition of process output streams that satisfy charge, material, momentum, and energy balances and equilibrium relationships for a given set of static process input streams. These first principle models have all of the equipment and process parameters needed for rigorous design or optimization of operating points. The transition between operating points is as fast as the convergence of the balances and relationships and consequently have no relationship to the time response of the process or automation system. The convergence of the models is a matter seconds for design but increases to minutes for RTO depending upon model scope and computing power.

Steady state models can be used to find the process gains that cause changes in the composition, density, pH, and temperature of process output streams for changes in the process input flows. However, levels are fixed and pressures do not take into account the effect of changing pressure drops with flow across piping resistances and control valves. Consequently, steady state models are not used to find the process gains for levels and pressures. Steady state models can be used as dynamic models of composition and temperature by the addition of dead time and filter blocks on the process outputs. The dead time block simulates the total loop dead time and the filter blocks simulates the primary and secondary time constants in the loop.

Dynamic models used for SAT, OTS, and PCI predict the changes in measured process variables (PV) for changes in manipulated variables (MV). Nearly all control systems end up affecting the process by manipulating a stream flow. Thus, the MV that directly affects the process is typically a control valve position or prime mover (e.g. compressor or pump) speed that changes a gas, liquid, or solid flow. A notable exception is the manipulation of electrical heater power to control the temperature in extruders, bench scale equipment for research, and crystal growth in semiconductor manufacturing.

Dynamic simulations use tieback, step response, multivariate statistical, neural network, and first principle models. Tieback models are predominantly used for SAT and OTS. Step response and first principle models are widely used for PCI. The PCI models can be used for OTS to significantly improve operator performance particularly for abnormal conditions and startups. However, there are often hundreds of parameters in first principle models with no general methodology for adjustment. Model Predictive Control potentially can be used to adjust parameters based on a model of the change in process variable for a change in the parameter. In this innovative application of MPC, the controlled variable is the model PV, the target is the actual process PV, and the manipulated variable is the model parameter. The MPC models can be identified offline by the running of the first principle dynamic models.

Multivariate statistical models are used for PCI of steady state processes and batch end points when the models have a large number of correlated inputs or unknown process relationships. Principal Component Analysis (PCA) is used to provide a reduced set of independent variables. Projection to Latent Structures (PLS) also known as Partial Least Squares provides a linear experimental model to predict a final process output using the independent inputs. Neural network models are used when the process relationships are extremely nonlinear and a piece wise linear fit is not practical. Neural network models can develop bizarre results for operating points outside of the range of process inputs used in the tests to develop the model. Neural Network models can also produce localized reversals of process gain from bumps or valleys in the prediction.

Multivariate statistical and neural network dynamic models presently rely on the insertion of dead time blocks on process inputs to model the effect on downstream process outputs. The lack of a process time constant or integrating response means that these models cannot be used for testing feedback control systems. For batch processes, the prediction of batch end point conditions does not require dead time blocks because there are no synchronization issues.

Whereas first principle steady state models zero out the ODE derivative, first principle dynamic models integrate the ODE derivative. The differential equations are largely the same except crystallization, evaporation, heat transfer, mass transfer, and reaction rates are used instead of equilibrium relationships to provide the dynamics between different operating conditions. The use of reaction kinetics and driving forces for heat and mass transfer rather than equilibriums is particularly important to show the process response during abnormal operation and startup. The differences in temperature across a surface with heat transfer coefficients and surface area are used as driving forces for heat transfer rates. Similarly, differences in composition across a surface with mass transfer coefficients and surface area are driving forces for mass transfer rates.  For modeling dissolved oxygen, the bubble surface area and the composition inside and outside the bubble is used in the driving force for mass transfer.

Dynamic first principle models should include the dynamics of the automation system besides the process as shown in Figure 12-1.  Valve and variable speed drive (VSD) models should include the installed flow characteristic, resolution and sensitivity limits, deadband, dead time, and a velocity limited exponential response. Measurement models should include transportation delays, sensor lag and delay, signal filtering, transmitter damping, resolution and sensitivity limits, and update delays. For analyzers, the measurement models should include sample transportation delays, cycle time, and multiplex time. Wireless devices should include the update rate and trigger level.

Figure 12-1: First Principle Dynamic Model (slide 1)

Step response models use an open loop gain, total loop dead time, and a primary and possibly a secondary time constant. The process gain is a steady state gain for self-regulating processes. The process gain is an integrating process gain for integrating and runaway processes. The inputs and outputs of the step response model are deviation variables. The input is a change in manipulated variable and the output is the change in the process variable. The models identified by tuner and MPC identification software take into account the controller scale ranges of the manipulated and process variables and include the effect of valve or variable speed drive and measurement dynamics. As a result the process gain identified is really an open loop gain that is the dimensionless product of the valve or VSD gain, process gain, and measurement gain. Correspondingly, the process dead time is actually a total loop dead time including the dynamics of the automation system. While the primary (largest) and secondary (second largest) time constants are normally in the process for composition and temperature control, they can be in the automation system for flow, level, and pressure control. Tieback models can be enhanced to use step response models.

Figure 12-2: Step Response Model (slide 2)

Standard tieback models pass the PID output through a multiplier block for the open loop gain and filter block for the primary time constant to create the PV input. The tieback inputs and outputs are typically in engineering units. Simple enhancements to this setup enables a step response models such as shown in Figure 12-2 to be used to provide a dynamic fidelity that is better than what can be achieved by first principle model whose parameters have not been adjusted based on test runs. Not shown are the limits to prevent values from exceeding scale ranges.

The enhancement to the input of standard tieback model is to subtract the normal operating value of the manipulated variable (%MVo) from the new value of the manipulated variable (%MVn) to create a deviation variable that is the change in manipulated variable. The enhancement to the output is to add the normal operating value of the process variable (%PVo) from the new value of the process variable (%PVn) to create a deviation variable  that is the change in process variable.  A dead time block for the total loop dead time and a filter block for the secondary time constant are inserted to provide all of the parameters for a second order plus dead time step response model for self- regulating processes. If the manipulated variable equals its normal operating point, the process variable will settle out to equal its normal operating point. The input and output biases enable the setting of normal operating points and the further enhancement of the tieback to use a step response model for higher fidelity simulations and linear dynamic estimators.

For integrating and runaway processes, an integrator block is substituted for the filter block for the primary time constant. For integrating process models, the normal operating point of the manipulated variable that is the negative bias must be sufficiently greater than zero to provide negative as well as positive changes in the process variable. This bias represents a process load. To achieve a new target or setpoint the manipulated variable must be temporarily changed to be different from the load. When the process variable settles out at the new operating point, the manipulated variable returns to be equal to the load.

Dynamic simulations for system acceptance testing (SAT), operator training systems (OTS), and process control improvement (PCI) use a virtual or actual version of the actual control system configuration and graphics, including historian and advanced control tools, interfaced to a dynamic model running in an application station or personal computer. The use of an actual or virtual rather than an emulated control system is necessary to allow the operators, process engineers, and automation engineers and technicians to use the same graphics, trends, and configuration as the actual installation. This fidelity to the actual installation is essential. The emulation of a PID block is problematic because of the numerous and powerful proprietary features, such as anti-reset windup and external-reset feedback. An actual control system is often used in SAT to include the testing of input and output channels. A virtual system is preferred for OTS and PCI to free up the control system hardware and enable speed-up and portability of the control system.

The fidelity of steady state models is the error between the final values of modeled and measured compositions, densities, flows, and temperatures. The fidelity of a dynamic model is judged not only by final values but also the time response. For PID control, the dead time and maximum excursion rate in the time frame of feedback correction is most important. For a PID tuned for good disturbance rejection, the time frame is 4 dead times.

Frist principle models can be speed up to be faster than real time by increasing the integration step size and the kinetic rates. The effect of these factors is multiplicative. Dynamic bioreactor models are run 1000 times real time by increasing the kinetics by a factor of 100 and the integration step size by a factor of 10. The result is a simulation batch cycle time of about 30 minutes rather than an actual batch time of two weeks. An increase in kinetics requires a proportional increase in the flows associated with the kinetics. The process time constants will be decreased by the speed up factor. Dead times should be proportionally decreased so that the time constant to dead time ratio is about the same enabling the PID controller gain in the virtual plant to be about the same as the PID gain used in the actual plant.  If a manipulated flow scale span is increased to account for an increase in kinetic rate, the  virtual PID gain should be decreased in proportion to the increase in the manipulated flow scale span. The PID rate and reset time should be decreased in proportion to the decrease in total loop dead time.

12.2 Linear Dynamic Estimators

Step response models can be used as linear dynamic estimators as shown in Figure 12-3 by using the steady state value of a process variable from Figure 12-2 as a future value after conversion from percent of scale to engineering units. The error between the model output that includes dead time and time constants (new PV value) and an at-line or off-line analyzer result (measured PV value) is used to bias the linear dynamic estimator output (future PV value) similar to the correction done in model predictive control (MPC). The linear dynamic estimator can be extended to include the step response models of several process inputs. MPC identification software can readily provide these models. Adaptive tuner software can provide the models by setting up dummy loops and generating tests from the step changes in the dummy controller manual or remote output. In the use of these step response models, the user must be aware despite the display of variables in engineering units these models are internally using values in percent of scale ranges because the controller algorithms are executed based on percent values of inputs and outputs. The dynamic estimator in Figure 12-3 is using variables in engineering units.

Figure 12-3: Linear Dynamic Estimator (slide 3)