This document discusses modeling and solving partial differential equations (PDEs) using the Industrial Modeling Framework (IMF). It presents a sample 1D heat transfer PDE problem and models it using finite difference (FD) and orthogonal collocation (OC) methods. The results show good agreement between the two methods. IMPL can model both dynamic and spatially distributed problems by discretizing across time and space to obtain algebraic equations that can then be optimized.

1. Partial Differential Equations (PDE’s)
Industrial Modeling Framework (PDE-IMF)
“Optimize Distributed Process Models in both Space and Time”
i n d u s t r IAL g o r i t h m s LLC. (IAL)
www.industrialgorithms.com
October 2014
Introduction to Partial Differential Equations (PDE’s), UOPSS and QLQP
Presented in this short document is a description of modeling and solving partial differential
equations (PDE’s) in both the temporal and spatial dimensions using IMPL. The sample PDE
problem is taken from Cutlip and Shacham (1999 and 2014) and models the process of
unsteady-state heat transfer or conduction in a one dimensional (1D) slab with one face
insulated and constant thermal conductivity as discussed by Geankoplis (1993). The linear
PDE which is first-order in time and second-order in space is described as follows:
∂T/∂t = a ∂2T/∂x2
“T” is the temperature phenomenon, “t” is the temporal dimension, “x” is the spatial dimension
(axial, horizontal, lateral, vertical, elevational, etc.) and “a” (2x10-5) is the thermal diffusivity. The
initial conditions have all temperatures starting in time at 100.0 except for the first temperature
which is externally controlled at 0.0 and is the first boundary condition. The second boundary
condition has an insulated boundary at the opposite side of the controlled temperature which
allows no heat conduction i.e., ∂T/∂x = 0.0.
For this small PDE problem we configure only one continuous-process unit-operation of subtype
blackbox called SLAB (see appendices A and B). There are no in- and out-port-states given
that there are no material flow exchanges. In IMPL, the temporal dimension is modeled
implicitly whereas the structural (UOPSS) and spatial (“x”) dimensions must be modeled
explicitly. For the spatial dimension, there are several viable methods to discretize space such
as the Method of Lines (MOL, Schiesser, 1991) and the Method of Weighted Residuals (MWR,
Villadsen and Michelsen, 1978). The first method is essentially finite-difference (FD) of space
and is similar to the Euler’s method IMPL uses to digitize time. The second method can use
orthogonal collocation (OC) which uses the roots of orthogonal polynomials to determine the
discretized spatial dimensions and is a popular, useful and easy-to-use method found in many
chemical engineering applications (Kelly, 1991) which can be applied with typically less spatial
points than FD with comparable accuracy.
Industrial Modeling Framework (IMF), IMPL and SSIIMPLE
To implement the mathematical formulation of this and other systems, IAL offers a unique
approach and is incorporated into our Industrial Modeling Programming Language we call IMPL.
IMPL 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, C#, VBA, Java (SWIG), Python (CTYPES)
and/or Julia (CCALL) 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

2. such as COINMP, GLPK, LPSOLVE, SCIP, CPLEX, GUROBI, LINDO, XPRESS, CONOPT,
IPOPT, KNITRO and WORHP 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. 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
(measurement and nodal/constraint tests) 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 IMPL is called SSIIMPLE (we hope literally) which is short
for Server, Solvers, Interfacer (IML), Interacter (IPL), Modeler, Presolver Libraries and
Executable. The Server, Solvers, Presolver and Executable are primarily model or problem-independent
whereas the Interfacer, Interacter 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, IMPL's standard
Interfacer, Interacter 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 IMPL 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 *.ILP 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 CIP,
LIP, SIP and KIP (constant, linear and monotonic spline interpolations) as well as user-written
extrinsic functions (XFCN). It is also possible to import another type of foreign file called the
*.INL file where both linear and nonlinear constraints can be added easily using new or existing
IMPL variables.
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 Gnome Dia and Matplotlib modules respectively but other
prototypes embedded within Microsoft Excel/VBA for example can be created in a
straightforward manner.

3. However, the primary purpose of the IMF's is to provide a timely, cost-effective, manageable
and maintainable deployment of IMPL to formulate and optimize complex industrial
manufacturing systems in either off-line or on-line environments. Using IMPL 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 IMPL 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.
Partial Differential Equations (PDE) Synopsis
At this point we explore further the application of modeling and solving PDE’s in IMPL
comparing FD with 11 (eleven) spatial points and OC with 5 (five) spatial or collocation points.
The problem is configured with 1-second time-period durations with a future time-horizon of
6,000 equidistant time-periods or seconds.
The FD formulation found in Appendix A is taken directly from Cutlip and Shachum (2014) and
our and their results found in Table 1 are very close. The problem is solved using IPOPT-MUMPS
in less than 1-second of CPU time with 1 (one) iteration and there are 60,001 equality
constraints, 60,001 variables and 227,992 Jacobian elements (with zero degrees-of-freedom).
Table 1. FD Temperatures at Time 6000-seconds.
Space Temperature
0 0.00E+00
0.1 1.62E+01
0.2 3.17E+01
0.3 4.60E+01
0.4 5.85E+01
0.5 6.90E+01
0.6 7.75E+01
0.7 8.38E+01
0.8 8.82E+01
0.9 9.08E+01
1 9.17E+01
The OC formulation found in Appendix B uses the Villadsen and Michelsen (1978) technique to
calculate the 5 (five) spatial collocation points (2 (two) exterior and 3 (three) interior) where the
first and second-order derivative weights as well as the interpolation weights are computed
using an external or extrinsic function (XFCN) coded in Fortran. The problem has 30,001
equality constraints, 30,001 variables and 143,998 Jacobian elements.

4. The results for the OC method are shown in Table 2 with the 5 (five) collocation points where x1
corresponds to x = 0.0 and x5 corresponds to x = 1.0. The row with x = 0.2 is an interpolated
value using the interpolation weights (C1, C2, …, C5) found in Appendix B.
Table 2. OC Temperatures at Time 6000-seconds.
Space Temperature
x1 0.00E+00
x2 1.79E+01
x3 6.88E+01
x4 9.09E+01
x5 9.18E+01
x = 0.2 3.12E+01
There is good agreement between the FD and OC methods for x5 (x = 1.0) and at x = 0.2 which
confirms that both methods are viable approaches to numerically solving PDE’s.
In summary, it should be clear that IMPL can be used to model and solve not only dynamic
problems but also “distributed” problems where the term distributed is used to describe
problems that are spatially distributed or dispersed as opposed to those that are “lumped” or
non-distributed. Essentially, these types of problems are simply discretized across not only time
but also space where ultimately algebraic equations result. Finally, once we have algebraic
equalities (and inequalities) then we can optimize instead of just simulating where we can either
maximize an economic objective function or minimize a parameter estimation sum of squares of
residuals to calibrate or fit model parameters.
References
Villadsen, J., Michelsen, M.L., “Solution of differential equation models by polynomial
approximation”, Prentice-Hall, (1978).
Schiesser, W.E., “The numerical model of lines”, Academic Press, (1991).
Kelly, J.D., “The design, construction, modeling, identification and multivariable constraint
control of a pilot plant fluidized bed catalytic reactor”, M.Eng., McMaster University, (1991).
Geankoplis, C.J., “Transport processes and unit operations”, 3rd Edition, Prentice-Hall, (1993).
Cutlip, M.B., Shacham, M., “Problem solving in chemical engineering with numerical methods,
Prentice-Hall, (1999).
Cutlip, M.B., Shacham, M.,”The numerical model of lines for partial differential equations”,
http://www.polymath-software.com/papers/cachen2.pdf, accessed October, (2014).
Appendix A – PDE-IMF-FD.IML File
i M P l (c)
Copyright and Property of i n d u s t r I A L g o r i t h m s LLC.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Calculation Data (Parameters)

5. !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sCalc,@sValue
PERIOD,1.0
START,-PERIOD
BEGIN,0.0
END,6000.0
dx,0.1
&sCalc,@sValue
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Chronological Data (Periods)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
@rPastTHD,@rFutureTHD,@rTPD
START,END,PERIOD
@rPastTHD,@rFutureTHD,@rTPD
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Construction Data (Pointers)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sUnit,&sOperation,@sType,@sSubtype,@sUse
SLAB,,processc,blackbox,,
&sUnit,&sOperation,@sType,@sSubtype,@sUse
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Condition Data (Properties)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sCondition
T1
T2
T3
T4
T5
T6
T7
T8
T9
T10
T11
eqT2
eqT3
eqT4
eqT5
eqT6
eqT7
eqT8
eqT9
eqT10
&sCondition
&sCoefficient,@sType,@sPath_Name,@sLibrary_Name,@sFunction_Name,@iNumber_Conditions,@rPerturb_Size,@sCondition_Names
a,static
&sCoefficient,@sType,@sPath_Name,@sLibrary_Name,@sFunction_Name,@iNumber_Conditions,@rPerturb_Size,@sCondition_Names
&sUnit,&sOperation,&sCondition,@rCondition_Lower,@rCondition_Upper,@rCondition_Target
SLAB,,T1,0.0,0.0,
SLAB,,T2,0.0,200.0,
SLAB,,T3,0.0,200.0,
SLAB,,T4,0.0,200.0,
SLAB,,T5,0.0,200.0,
SLAB,,T6,0.0,200.0,
SLAB,,T7,0.0,200.0,
SLAB,,T8,0.0,200.0,
SLAB,,T9,0.0,200.0,
SLAB,,T10,0.0,200.0,
SLAB,,T11,0.0,200.0,
SLAB,,eqT2,0.0,0.0,
SLAB,,eqT3,0.0,0.0,
SLAB,,eqT4,0.0,0.0,
SLAB,,eqT5,0.0,0.0,
SLAB,,eqT6,0.0,0.0,
SLAB,,eqT7,0.0,0.0,
SLAB,,eqT8,0.0,0.0,
SLAB,,eqT9,0.0,0.0,
SLAB,,eqT10,0.0,0.0,
&sUnit,&sOperation,&sCondition,@rCondition_Lower,@rCondition_Upper,@rCondition_Target
&sUnit,&sOperation,&sCoefficient,@rCoefficient_Lower,@rCoefficient_Upper,@rCoefficient_Target
SLAB,,a,2.0E-5,2.0E-5,
&sUnit,&sOperation,&sCoefficient,@rCoefficient_Lower,@rCoefficient_Upper,@rCoefficient_Target
ConditionsUOCondition-&sUnit,&sOperation,&sCondition,@sType,@rValue,@sValue
SLAB,,eqT2,?,3,T2 - T2[-1] - a / dx^2.0 * (T3-2.0*T2+T1) * PERIOD
SLAB,,eqT3,?,3,T3 - T3[-1] - a / dx^2.0 * (T4-2.0*T3+T2) * PERIOD
SLAB,,eqT4,?,3,T4 - T4[-1] - a / dx^2.0 * (T5-2.0*T4+T3) * PERIOD
SLAB,,eqT5,?,3,T5 - T5[-1] - a / dx^2.0 * (T6-2.0*T5+T4) * PERIOD
SLAB,,eqT6,?,3,T6 - T6[-1] - a / dx^2.0 * (T7-2.0*T6+T5) * PERIOD
SLAB,,eqT7,?,3,T7 - T7[-1] - a / dx^2.0 * (T8-2.0*T7+T6) * PERIOD
SLAB,,eqT8,?,3,T8 - T8[-1] - a / dx^2.0 * (T9-2.0*T8+T7) * PERIOD

6. SLAB,,eqT9,?,3,T9 - T9[-1] - a / dx^2.0 * (T10-2.0*T9+T8) * PERIOD
SLAB,,eqT10,?,3,T10 - T10[-1] - a / dx^2.0 * (T11-2.0*T10+T9) * PERIOD
SLAB,,T11,?,3,(4.0 * T10 - T9) / 3.0
ConditionsUOCondition-&sUnit,&sOperation,&sCondition,@sType,@rValue,@sValue
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Cost Data (Pricing)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sUnit,&sOperation,&sCondition,@rConditionPro_Weight,@rConditionPer1_Weight,@rConditionPer2_Weight,@rConditionPen_Weight
&sUnit,&sOperation,&sCondition,@rConditionPro_Weight,@rConditionPer1_Weight,@rConditionPer2_Weight,@rConditionPen_Weight
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Content Data (Past, Present Provisos)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sUnit,&sOperation,&sCondition,@rCondition_Value,@rStart_Time
SLAB,,T2,100.0,0.0
SLAB,,T3,100.0,0.0
SLAB,,T4,100.0,0.0
SLAB,,T5,100.0,0.0
SLAB,,T6,100.0,0.0
SLAB,,T7,100.0,0.0
SLAB,,T8,100.0,0.0
SLAB,,T9,100.0,0.0
SLAB,,T10,100.0,0.0
&sUnit,&sOperation,&sCondition,@rCondition_Value,@rStart_Time
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Command Data (Future Provisos)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sUnit,&sOperation,@rSetup_Lower,@rSetup_Upper,@rBegin_Time,@rEnd_Time
SLAB,,1,1,BEGIN,END
&sUnit,&sOperation,@rSetup_Lower,@rSetup_Upper,@rBegin_Time,@rEnd_Time
Appendix B – PDE-IMF-OC.IML File
i M P l (c)
Copyright and Property of i n d u s t r I A L g o r i t h m s LLC.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Calculation Data (Parameters)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sCalc,@sValue
PERIOD,1.0
START,-PERIOD
BEGIN,0.0
END,6000.0
&sCalc,@sValue
XFCN-@sPath_Name,@sLibrary_Name,@sFunction_Name
C:IndustrialAlgorithmsProceduresx64Release,xfunc_ocl.dll,xfunc_ocl
XFCN-@sPath_Name,@sLibrary_Name,@sFunction_Name
&sCalc,@sValue
ALPHA,0.0
BETA,0.0
A11,XFCN(1;1;3;1;1;1;ALPHA;BETA)
A12,XFCN(1;2;3;1;1;1;ALPHA;BETA)
A13,XFCN(1;3;3;1;1;1;ALPHA;BETA)
A14,XFCN(1;4;3;1;1;1;ALPHA;BETA)
A15,XFCN(1;5;3;1;1;1;ALPHA;BETA)
A21,XFCN(2;1;3;1;1;1;ALPHA;BETA)
A22,XFCN(2;2;3;1;1;1;ALPHA;BETA)
A23,XFCN(2;3;3;1;1;1;ALPHA;BETA)
A24,XFCN(2;4;3;1;1;1;ALPHA;BETA)
A25,XFCN(2;5;3;1;1;1;ALPHA;BETA)
A31,XFCN(3;1;3;1;1;1;ALPHA;BETA)
A32,XFCN(3;2;3;1;1;1;ALPHA;BETA)
A33,XFCN(3;3;3;1;1;1;ALPHA;BETA)
A34,XFCN(3;4;3;1;1;1;ALPHA;BETA)
A35,XFCN(3;5;3;1;1;1;ALPHA;BETA)
A41,XFCN(4;1;3;1;1;1;ALPHA;BETA)
A42,XFCN(4;2;3;1;1;1;ALPHA;BETA)
A43,XFCN(4;3;3;1;1;1;ALPHA;BETA)
A44,XFCN(4;4;3;1;1;1;ALPHA;BETA)
A45,XFCN(4;5;3;1;1;1;ALPHA;BETA)
A51,XFCN(5;1;3;1;1;1;ALPHA;BETA)
A52,XFCN(5;2;3;1;1;1;ALPHA;BETA)
A53,XFCN(5;3;3;1;1;1;ALPHA;BETA)
A54,XFCN(5;4;3;1;1;1;ALPHA;BETA)
A55,XFCN(5;5;3;1;1;1;ALPHA;BETA)
B11,XFCN(1;1;3;1;1;2;ALPHA;BETA)
B12,XFCN(1;2;3;1;1;2;ALPHA;BETA)
B13,XFCN(1;3;3;1;1;2;ALPHA;BETA)
B14,XFCN(1;4;3;1;1;2;ALPHA;BETA)
B15,XFCN(1;5;3;1;1;2;ALPHA;BETA)
B21,XFCN(2;1;3;1;1;2;ALPHA;BETA)

7. B22,XFCN(2;2;3;1;1;2;ALPHA;BETA)
B23,XFCN(2;3;3;1;1;2;ALPHA;BETA)
B24,XFCN(2;4;3;1;1;2;ALPHA;BETA)
B25,XFCN(2;5;3;1;1;2;ALPHA;BETA)
B31,XFCN(3;1;3;1;1;2;ALPHA;BETA)
B32,XFCN(3;2;3;1;1;2;ALPHA;BETA)
B33,XFCN(3;3;3;1;1;2;ALPHA;BETA)
B34,XFCN(3;4;3;1;1;2;ALPHA;BETA)
B35,XFCN(3;5;3;1;1;2;ALPHA;BETA)
B41,XFCN(4;1;3;1;1;2;ALPHA;BETA)
B42,XFCN(4;2;3;1;1;2;ALPHA;BETA)
B43,XFCN(4;3;3;1;1;2;ALPHA;BETA)
B44,XFCN(4;4;3;1;1;2;ALPHA;BETA)
B45,XFCN(4;5;3;1;1;2;ALPHA;BETA)
B51,XFCN(5;1;3;1;1;2;ALPHA;BETA)
B52,XFCN(5;2;3;1;1;2;ALPHA;BETA)
B53,XFCN(5;3;3;1;1;2;ALPHA;BETA)
B54,XFCN(5;4;3;1;1;2;ALPHA;BETA)
B55,XFCN(5;5;3;1;1;2;ALPHA;BETA)
C1,XFCN(0.2;1;3;1;1;ALPHA;BETA)
C2,XFCN(0.2;2;3;1;1;ALPHA;BETA)
C3,XFCN(0.2;3;3;1;1;ALPHA;BETA)
C4,XFCN(0.2;4;3;1;1;ALPHA;BETA)
C5,XFCN(0.2;5;3;1;1;ALPHA;BETA)
&sCalc,@sValue
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Chronological Data (Periods)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
@rPastTHD,@rFutureTHD,@rTPD
START,END,PERIOD
@rPastTHD,@rFutureTHD,@rTPD
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Construction Data (Pointers)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sUnit,&sOperation,@sType,@sSubtype,@sUse
SLAB,,processc,blackbox,,
&sUnit,&sOperation,@sType,@sSubtype,@sUse
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Condition Data (Properties)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sCondition
T1
T2
T3
T4
T5
TX
eqT2
eqT3
eqT4
eqT5
eqTX
&sCondition
&sCoefficient,@sType,@sPath_Name,@sLibrary_Name,@sFunction_Name,@iNumber_Conditions,@rPerturb_Size,@sCondition_Names
a,static
&sCoefficient,@sType,@sPath_Name,@sLibrary_Name,@sFunction_Name,@iNumber_Conditions,@rPerturb_Size,@sCondition_Names
&sUnit,&sOperation,&sCondition,@rCondition_Lower,@rCondition_Upper,@rCondition_Target
SLAB,,T1,0.0,0.0,
SLAB,,T2,0.0,200.0,
SLAB,,T3,0.0,200.0,
SLAB,,T4,0.0,200.0,
SLAB,,T5,0.0,200.0,
SLAB,,TX,0.0,200.0,
SLAB,,eqT2,0.0,0.0,
SLAB,,eqT3,0.0,0.0,
SLAB,,eqT4,0.0,0.0,
SLAB,,eqT5,0.0,0.0,
&sUnit,&sOperation,&sCondition,@rCondition_Lower,@rCondition_Upper,@rCondition_Target
&sUnit,&sOperation,&sCoefficient,@rCoefficient_Lower,@rCoefficient_Upper,@rCoefficient_Target
SLAB,,a,2.0E-5,2.0E-5,
&sUnit,&sOperation,&sCoefficient,@rCoefficient_Lower,@rCoefficient_Upper,@rCoefficient_Target
ConditionsUOCondition-&sUnit,&sOperation,&sCondition,@sType,@rValue,@sValue
SLAB,,eqT2,?,3,T2 - T2[-1] - a * (B21*T1+B22*T2+B23*T3+B24*T4+B25*T5) * PERIOD
SLAB,,eqT3,?,3,T3 - T3[-1] - a * (B31*T1+B32*T2+B33*T3+B34*T4+B35*T5) * PERIOD
SLAB,,eqT4,?,3,T4 - T4[-1] - a * (B41*T1+B42*T2+B43*T3+B44*T4+B45*T5) * PERIOD
SLAB,,eqT5,?,3,(A51*T1+A52*T2+A53*T3+A54*T4+A55*T5)
SLAB,,TX,?,3,(C1*T1+C2*T2+C3*T3+C4*T4+C5*T5)
ConditionsUOCondition-&sUnit,&sOperation,&sCondition,@sType,@rValue,@sValue
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Cost Data (Pricing)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

8. &sUnit,&sOperation,&sCondition,@rConditionPro_Weight,@rConditionPer1_Weight,@rConditionPer2_Weight,@rConditionPen_Weight
&sUnit,&sOperation,&sCondition,@rConditionPro_Weight,@rConditionPer1_Weight,@rConditionPer2_Weight,@rConditionPen_Weight
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Content Data (Past, Present Provisos)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sUnit,&sOperation,&sCondition,@rCondition_Value,@rStart_Time
SLAB,,T2,100.0,0.0
SLAB,,T3,100.0,0.0
SLAB,,T4,100.0,0.0
&sUnit,&sOperation,&sCondition,@rCondition_Value,@rStart_Time
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Command Data (Future Provisos)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
&sUnit,&sOperation,@rSetup_Lower,@rSetup_Upper,@rBegin_Time,@rEnd_Time
SLAB,,1,1,BEGIN,END
&sUnit,&sOperation,@rSetup_Lower,@rSetup_Upper,@rBegin_Time,@rEnd_Time