Advanced Process Monitoring Industrial Modeling Framework (APM-IMF)

i n d u s t r IAL g o r i t h m s LLC. (IAL)

www.industrialgorithms.com July 2013

Introduction to Advanced Process Monitoring, UOPSS and QLQP Presented in this short document is a description of what is called Advanced Process Monitoring (APM) as described by Hedengren (2013). APM is the term given to the technique of estimating unmeasured but observable variables or "states" using statistical data reconciliation and regression (DRR) in an off-line or real-time environment and is also referred to as Moving Horizon Estimation (MHE) (Robertson et. al., 1996). Essentially, the model and data define a simultaneous nonlinear and dynamic DRR problem where the model is either engineering-based (first-principles, fundamental, mechanistic, causal, rigorous) or empirical-based (correlation, statistical data-based, observational, regressed) or some combination of both (hybrid). Figure 1 depicts a very simple flowsheet problem with a continuously-stirred tank reactor (CSTR) configured as a black-box in our unit-operation-port-state superstructure (UOPSS) (Kelly, 2004a, 2005, and Zyngier and Kelly, 2012) taken from Henson and Seborg (1997).

Figure 1. Simple CSTR Problem with One Reaction of Component A to B.

The single rectangular box with the cross-hairs is a "continuous-process" type of unit-operation which can be of the "black-box" subtype and for this simple model has no in or out-ports given that there is no explicit exchange of substances or resources. Black-boxes allow adhoc, custom or user formula to be configured for the unit-operation where the variables referenced inside the expressions are what we call "conditions". Each formula can be any nonlinear function of conditions and "coefficients". Conditions represent operating or processing conditions such as severity, temperature, pressure, etc. where coefficients refer to reaction frequency factors, catalyst activities, activation energies, heat capacities, gas constants, etc. or other physical, thermodynamic, hydraulic or kinetic properties and can be either fixed or variable. If variable, then these may estimated or fitted from the data similar to parameter estimation or weighted least-squares regression although we extend this to also use data reconciliation which is identical to Error-in-Variables (EiV) regression (Kelly, 1998). For this small CSTR example, there is one chemical reaction of component A to B where the nomenclature is found in Figure 1. This model is both dynamic and nonlinear where the concentration of A is found in equation (1) in difference form and the reactor temperature is found in difference equation (2) below. V * Ca[0] = V * Ca[-1] + q * dt * (Caf - Ca[0]) - V * k0 * dt * Ca[0] * EXP(-EoverR / T) (1) rho * Cp * V * T[0] = rho * Cp * V * T[-1] + q * dt * rho * Cp * (Tf - T[0]) + k0 * dt * V * mdelH * Ca * EXP(-EoverR / T[0]) + UA * dt * (Tc - T[0]) (2) where "dt" is the time-period duration or sampling interval and [0] and [-1] are the time lags or delays in the present and past. We use simple Euler's method to convert the ordinary differential equations into simple algebraic equations that can be solved using optimization techniques. Euler's method is very practical and effective where others may use orthogonal or spline collocation. Although collocation on finite elements is more parsimonious, Euler's method is easy to understand and implement and when used with sparse matrix solvers can be numerically competitive as well. These types of engineering equations are also known as "lumped" models where "distributed" models include spatial, axial, radial, longitudinal, etc. independent dimensions other than time which can be similarly converted to algebraic form using collocation on finite elements. Industrial Modeling Framework (IMF), IMPRESS and SIIMPLE To implement the mathematical formulation of this and other systems, IAL offers a unique approach and is incorporated into our Industrial Modeling and Pre-Solving System we call IMPRESS. IMPRESS has its own modeling language called IML (short for Industrial Modeling Language) which is a flat or text-file interface as well as a set of API's which can be called from any computer programming language such as C, C++, Fortran, Java (SWIG), C# or Python (CTYPES) called IPL (short for Industrial Programming Language) to both build the model and to view the solution. Models can be a mix of linear, mixed-integer and nonlinear variables and constraints and are solved using a combination of LP, QP, MILP and NLP solvers such as COINMP, GLPK, LPSOLVE, SCIP, CPLEX, GUROBI, LINDO, XPRESS, CONOPT, IPOPT and KNITRO as well as our own implementation of SLP called SLPQPE (Successive Linear & Quadratic Programming Engine) which is a very competitive alternative to the other nonlinear solvers and embeds all available LP and QP solvers. In addition and specific to DRR problems, we also have a special solver called SECQPE standing for Sequential Equality-Constrained QP Engine which computes the least-squares

solution and a post-solver called SORVE standing for Supplemental Observability, Redundancy and Variability Estimator to estimate the usual DRR statistics found in Kelly (1998 and 2004a) and Kelly and Zyngier (2008). SECQPE also includes a Levenberg-Marquardt regularization method for nonlinear data regression problems and can be presolved using SLPQPE i.e., SLPQPE warm-starts SECQPE. SORVE is run after the SECQPE solver and also computes the well-known "maximum-power" gross-error statistics to help locate outliers, defects and/or faults i.e., mal-functions in the measurement system and mis-specifications in the logging system. The underlying system architecture of IMPRESS is called SIIMPLE (we hope literally) which is short for Server, Interacter (IPL), Interfacer (IML), Modeler, Presolver Libraries and Executable. The Server, Presolver and Executable are primarily model or problem-independent whereas the Interacter, Interfacer and Modeler are typically domain-specific i.e., model or problem-dependent. Fortunately, for most industrial planning, scheduling, optimization, control and monitoring problems found in the process industries, IMPRESS's standard Interacter, Interfacer and Modeler are well-suited and comprehensive to model the most difficult of production and process complexities allowing for the formulations of straightforward coefficient equations, ubiquitous conservation laws, rigorous constitutive relations, empirical correlative expressions and other necessary side constraints. User, custom, adhoc or external constraints can be augmented or appended to IMPRESS when necessary in several ways. For MILP or logistics problems we offer user-defined constraints configurable from the IML file or the IPL code where the variables and constraints are referenced using unit-operation-port-state names and the quantity-logic variable types. It is also possible to import a foreign LP file (row-based MPS file) which can be generated by any algebraic modeling language or matrix generator. This file is read just prior to generating the matrix and before exporting to the LP, QP or MILP solver. For NLP or quality problems we offer user-defined formula configuration in the IML file and single-value and multi-value function blocks writable in C, C++ or Fortran. The nonlinear formulas may include intrinsic functions such as EXP, LN, LOG, SIN, COS, TAN, MIN, MAX, IF, NOT, EQ, NE, LE, LT, GE, GT and KIP, LIP, SIP (constant, linear and monotonic spline interpolation) as well as user-written extrinsic functions. Industrial modeling frameworks or IMF's are intended to provide a jump-start to an industrial project implementation i.e., a pre-project if you will, whereby pre-configured IML files and/or IPL code are available specific to your problem at hand. The IML files and/or IPL code can be easily enhanced, extended, customized, modified, etc. to meet the diverse needs of your project and as it evolves over time and use. IMF's also provide graphical user interface prototypes for drawing the flowsheet as in Figure 1 and typical Gantt charts and trend plots to view the solution of quantity, logic and quality time-profiles. Current developments use Python 2.3 and 2.7 integrated with open-source Dia and Matplotlib modules respectively but other prototypes embedded within Microsoft Excel/VBA for example can be created in a straightforward manner. However, the primary purpose of the IMF's is to provide a timely, cost-effective, manageable and maintainable deployment of IMPRESS to formulate and optimize complex industrial manufacturing systems in either off-line or on-line environments. Using IMPRESS alone would be somewhat similar (but not as bad) to learning the syntax and semantics of an AML as well as having to code all of the necessary mathematical representations of the problem including the details of digitizing your data into time-points and periods, demarcating past, present and future time-horizons, defining sets, index-sets, compound-sets to traverse the network or topology, calculating independent and dependent parameters to be used as coefficients and bounds and

finally creating all of the necessary variables and constraints to model the complex details of logistics and quality industrial optimization problems. Instead, IMF's and IMPRESS provide, in our opinion, a more elegant and structured approach to industrial modeling and solving so that you can capture the benefits of advanced decision-making faster, better and cheaper. Advanced Process Monitoring Synopsis At this point we explore further the purpose of advanced process monitoring in terms of its estimation and diagnostic capability. For this small CSTR example we focus our attention on estimating the value and variance of the heat transfer coefficient times the reactor cooling jacket's surface area (UA) for a snap-shot of time within its dynamic operation. The rationale for this is to institute a monitoring program to periodically and accurately check the UA coefficient and to determine if it is deteriorating to a point where the cooling jacket needs to cleaned and/or back-flushed. We have chosen to highlight a single snap-shot of simulated processing data with a 30-second time-horizon and a sampling duration of 0.1-seconds resulting in 300 equally-spaced time-periods. The temporal data was generated in IMPRESS by using the nominal UA value of 50,000 J/K/s and arbitrarily perturbing the jacket cooling temperature (Tc), feed flow (q), feed temperature (Tf) and feed concentration (Caf) variables as shown in Figure 2 whilst recording the reactor concentration (Ca) and its temperature (T) over time with no measurement or process noise superimposed. For all other coefficients, default values are taken from Henson and Seborg (1997).

Figure 2. Condition Time Profiles for a 30-second Horizon with 0.1-second Intervals.

To estimate the UA coefficient, both the independent and dependent variable profiles are inputted into IMPRESS where the independent variable values are fixed and the dependent variable values are minimized according to a weighted sum of squares objective function found in any DRR problem. The raw standard deviations used as weights (i.e., weight = 1 / variance) for the reactor concentration and temperature are 0.01 mol/m3 and 1.0 K respectively. Given that no random nor systemic error was added to the data set, the DRR objective function is

2.1x10^(-5) and the value and variance for the single UA parameter is 50,000.1 and 25,164.1 respectively. This translates into a standard error of SQRT(25,164.1) = 158.6 and using an estimate of 2.0 for the Student-t statistic at 95% confidence yields an interval of 49,682.9 to 50,317.3 indicating that the UA parameter is estimated significantly. The SECQPE solver required approximately 9 iterations which is typical of most DRR problems without appreciable gross-errors and solved in less than 1-second. In summary and albeit a hypothetical example, this demonstrates that time-varying data can be used along with a nonlinear and dynamic model to fit observable parameters such as the heat transfer coefficient times area (UA). Performing APM either separately on selected equipment or simultaneously on several units is also possible and serves as a best practice approach to monitoring your production or process on a regular basis. References Robertson, D.G., Lee, J.H., Rawlings, J.B., "A moving horizon-based approach for least-squares estimation", American Institute of Chemical Engineering Journal, 42, 2209, (1996) Henson, M.A., Seborg, D.E., "Nonlinear process control", Prentice Hall, New Jersey, (1997). Kelly, J.D., "A regularization approach to the reconciliation of constrained data sets", Computers & Chemical Engineering, 1771, (1998). Kelly, J.D., "Production modeling for multimodal operations", Chemical Engineering Progress, February, 44, (2004a). Kelly, J.D., "Techniques for solving industrial nonlinear data reconciliation problems", Computers & Chemical Engineering, 2837, (2004b). Kelly, J.D., Mann, J.L., Schulz, F.G., "Improve accuracy of tracing production qualities using successive reconciliation", Hydrocarbon Processing, April, (2005). Kelly, J.D., "The unit-operation-stock superstructure (UOSS) and the quantity-logic-quality paradigm (QLQP) for production scheduling in the process industries", In: MISTA 2005 Conference Proceedings, 327, (2005). Kelly, J.D., Zyngier, D., "A new and improved MILP formulation to optimize observability, redundancy and precision for sensor network problems", American Institute of Chemical Engineering Journal, 54, 1282, (2008). Zyngier, D., Kelly, J.D., "UOPSS: a new paradigm for modeling production planning and scheduling systems", ESCAPE 22, June, (2012). Hedengren, J.D., "Advanced Process Monitoring", http://apm.byu.edu/pubs/springer.pdf (2013). Appendix A - APM-IMF.IML File