Dynamische Simulatoren in der Reaktorentwicklung. Ein Vergleich

Complete English translation of the original German-language document (51 pages).

Nuclear Research Center Karlsruhe

January 1969 — External Report 8/69-1

Institute for Reactor Development Data Processing Center

Dynamic Simulators in Reactor Development. A Comparison

W. Frisch G. Wilhelmi

Gesellschaft für Kernforschung mbH., Karlsruhe

[page 3: duplicate title page]

Section	Title	Page
1.	Introduction	1
2.	Dynamic Simulators	1
2.1	Criteria for Evaluating Simulators	1
2.2	Requirements Imposed	3
2.3	Development Trends in Simulators	4
2.4	Selected Simulators	8
3.	Problem Analysis	10
3.1	Dynamic Problems in Reactor Development	10
3.2	Solution Methods	11
3.3	Properties of the Systems of Equations	13
3.4	Test Examples	17
4.	System Comparison	20
4.1	System Properties	20
4.2	Process Preparation	26
4.3	Process Execution	28
5.	Future Development	29
5.1	Hardware	29
5.2	Software	30
5.3	Possible Increase in Computing Speed	31
6.	Summary	35
	References	39

1. Introduction

In the present comparison study, several dynamic simulators are examined with respect to their suitability for solving dynamic problems in reactor development. The comparison is motivated by the fact that the Data Processing Center (Datenverarbeitungszentrale) of the Institute for Reactor Development already operates 2 digital simulators. Since this has led to the observation that both simulators are not equally well suited for all problems encountered in reactor development, and because additionally a HYBRID-Programming system including the FORTRAN IV language is available, it appeared appropriate to examine all 3 systems and also 4 selected digital simulators with respect to their suitability for the problems at hand.

It should be noted that this is a practical comparison rather than a theoretical one. The suitability of each system is judged not only by its properties but also — and equally — by taking into account the practical conditions (computational effort, programming expenditure, turnaround time). The systems assessed are described in Sec. 2.4. Qualitative assessments are made possible by test examples derived from reactor development problems and analyzed in Sec. 3.4.

2. Dynamic Simulators

2.1 Criteria for Evaluating Simulators

A dynamic simulator can best be defined by distinguishing it from other computational systems. An analog simulator is a physical system in which — in analogy to the actual system — physical processes take place (e.g., electrical). The variables of the analog system correspond to those of the actual system and the analog has a structural similarity with the real system.

A digital simulator solves the differential equations that describe the system, i.e., it executes numerical integration processes. Simulators that work in real time are sometimes also referred to as simulators in the narrower sense.

Simulators for which one seeks to express the suitability (Eignung) for a given problem by means of a numerical value form a special case. To judge the suitability of a given simulator for a given problem, the following criteria can be established:

1. Solution Quality

a) Accuracy of Results b) Reproducibility with Different Loads c) Numerical Quality

2. Operational Convenience

a) Programming Effort b) Availability of a Problem-Oriented Programming Language c) Ease of Use

3. Processing Speed

a) Execution of the Linear Calculation b) Support for the Iterative Solution

2.2 Required Information

Based on the criteria for evaluating dynamic simulators, the following items of information are needed about the system, which are also important for the practical application of the simulator, and which make possible a comprehensive assessment (comparison) of the system:

All physically and mathematically correct problems should be solvable, and with acceptable computational effort it should be possible to specify the input data in a simple form.
The user should not be forced to modify the hardware structure and should always be at liberty to specify all necessary input information. It should be possible to provide this information and to modify the problem description easily, e.g., through corresponding input procedures. Programming of iterations should also be possible.
The Processing and Control of the computation steps, i.e., by means of automatic step-width control, error checking during the computation, output functions, etc.
The sequence should as far as possible take place automatically (without hardware intervention) and also allow output control (e.g., output of intermediate results and graphical output). The sequence should also permit an automatic comparison of the running values with specified limit values.
The comparison and restart operations (repeated integration over particular time intervals) are important.

No single simulator can meet all practical requirements and attain the greatest possible accuracy for every problem. The general practical comparison is therefore not capable of definitively determining which simulator is the most advantageous; the decision depends very much on the problem.

2.3 Development Trends in Simulators

Given the requirement for the processing of problems in reactor development, it is important to establish what information is contained in the simulator. This is where the differences between simulators are most apparent.

Through all existing and technically developed methods, the goal is to achieve a system that, while being as far as possible problem-neutral, at the same time is capable of being used immediately by the engineer without an extensive mathematical preliminary examination. This should be achieved for all common problems without special effort; the simulator should be capable of providing a correct and reliable solution with sufficient accuracy.

The digital approach to a dynamic simulator places the following demands on the hardware in order to fulfill the general requirements:

It must be possible to handle the entire problem-preparation stage (hardware setup, potentiometer setting) in digital form.
System variations should be possible without hardware intervention.
Iteration of the time integration must be possible by means of programming functions (Loop programming, FORTRAN, MIMIC).

Apart from the already established hardware functions, the development of digital simulators is heading toward the following:

Automation of problem preparation (function tables, patching, parameter setting).
Automation of the programming language (free-form language like FORTRAN, MIMIC, CSSL).
Synthesis of ideas in a given context or condition that is likely to offer more reliable solutions.
Building in greater speed of computation at comparable cost and with the aid of improved integration methods.
Enabling faster computation by distributing the computational work to several sub-processors (Multi-Processor system).

Such a development includes, from the hardware side, the increasing dominance of the digital simulator over the analog simulator, because the digital simulator offers the following advantages:

Numerical calculation: An arbitrary accuracy of results is attainable (in principle), even when the system itself has electronic disturbances or inaccuracies. Controllable numerical quality (stability, truncation errors).
Programming: There is no hardware problem to solve when a problem structure changes. Operations are freely programmable; iterations can be programmed.
The problem can be put down in writing in a problem-oriented language at any time, along with the documentation.
Automatic output in tabular form. The integration results can be graphically output directly.
Connection to large-scale computers via data communication (in the case of a separate digital simulator) or use of the available large-scale computer installation.

This development simultaneously requires, from the software side, the following improvements in the digital simulator over existing systems:

Simulation is linked to or performed on the digital simulator with a real number; an output in the form of a phase-plane representation should be achievable.
Simulation should, on the one hand, be rapid and, on the other hand, should also be sufficiently accurate.

2.3.1 Hybrid Simulators

A further possibility, mentioned here chiefly in connection with the digital computer, is the hybrid simulator. The digital computer carries out, in the hybrid simulator, all functions that require high accuracy or great flexibility, while the analog computer handles all operations that are to be executed with high speed. The digital machine also takes on the control of the entire computation process and the output of results.

The following enumeration shows the distribution of functions which is generally found in a hybrid simulator:

Execution of the linear part of the simulation task on the digital side (parameter storage, function generation, computation of linear equations).
System variations take place either on the parallel analog or on the digital simulator (digital-analog conversion, potentiometer setting programs, FORTRAN).
Execution of the Iteration programs with individual function steps (limiting functions, programs, MIMIC).

Apart from the already established hardware functions, one can also see a connection from the hybrid side: the increasing dominance of the digital simulator over the analog machine because the digital simulator carries out all tasks for which the analog machine is not well suited (see §7).

2.3.2 Digital Simulators

There is a broader interest in digital simulators, for whereas they come into consideration for the solution of dynamic problems as an independent tool and also with combined programming languages (FORTRAN, MIMIC, CSSL), they also make possible programming and solution of many particular problems that previously could only be treated by other means.

Concerning the hardware side, a digital simulator places the following demands in order to satisfy the general requirements:

As far as possible, the entire problem preparation (hardware programming, parameter setting) must be carried out in digital form.
System variations shall be possible by means of programming and without hardware intervention.
Iteration of the time integration shall be programmable by means of programming functions.

Apart from hardware functions already standardized, the development of digital simulators is heading in the following directions:

Automation of problem preparation (function tables, patching, parameter setting).
Automation of the programming language (free-format languages like FORTRAN, MIMIC, CSSL).
Greater speed of computation by improved integration methods.
Greater speed of computation by distributing the computational load among several sub-processors (Multi-Processor arrangement).
Greater speed of computation by using simulation languages that make it possible to split the computation into parallel branches.

2.3.3 Selected Simulators

For the comparison described in Sec. 4, a total of 3 simulators were selected, of which the first is the most widely applicable (IBM-, Filter-, Flip-program), register-based digital simulator at 256 integrators. Additionally, 2 digital simulators are described.

The static computational speed of all three is approximately 10^5 multiplications/second. The digital simulators achieve this through hardware acceleration of the arithmetic (fast floating-point operations). The system (IBM) achieved this performance through optimized compiler use. The functional characteristics of the selected systems are described in Sec. 4. The hardware equipment and the software functions were taken into account in the comparison.

2.4.1 TRICE — EAI

The system EAI 640 is a digital-hybrid simulator, but its characteristic feature is that it can most simply be characterized as a digital simulator. It permits a complete simulation from the computation standpoint and therefore is in many respects a worthwhile alternative to the classic hybrid simulator. It also opens new possibilities for the simulation of problems, for example a phase-plane representation can be produced without difficulty. The problem preparation for this system takes place in a special patching language, which makes possible the direct transfer of the block diagram. The system was presented at the last Joint Computer Conference in the autumn of 1968.

2.4.2 IBM 360/75S

IBM 360/75S is a digital simulator. For this system, a special digital simulator (CSMP) has been developed that handles, as a simulation language, the requirements of dynamic simulation. The system is configured so that problems in the CSMP language (Continuous System Modeling Program) can be solved in connection with the FORTRAN language on the IBM 360/75 computer. The IBM CSMP includes all dynamic operations necessary for processing dynamic problems:

Simple integration
Re-entry (restart operations)
Output function (tabular and graphical output)
Parameter variations (parameter studies)

The problem preparation for this system takes place in CSMP language. The system was presented in 1967 and is used in several research centers. It comes into consideration as an independent system for the processing of dynamic problems.

3. Problem Analysis

3.1 Dynamic Problems in Reactor Development

The simulation of dynamic processes is possible with all types of simulators, since the basic task of every simulator is to solve dynamical problems numerically or by means of an analog computation. The goal is to reduce a problem to a form that can be solved by the simulator and to achieve the most rapid and accurate solution possible.

The most typical dynamic problems in reactor development that come into consideration include the following:

Thermodynamics of the Reactor
Reactor Control
Neutron Kinetics
Shutdown Systems (e.g., control rod systems)
Material Transport (e.g., of fuel elements)
Coolant Circulation
Stability of the Coolant Flow
Criticality Calculations
Reactor Dynamics

For each of these dynamic problems, an appropriate simulator or computation method can and should be found and used. The comparison considers how well the selected simulators are suited for each of these problem classes.

3.2 Solution Methods

The simulation programs available fall into 4 groups, namely:

a) Analytical programs b) Conformal (Transformation) methods c) Statistical methods d) Simulation of the Actual System

To a)

Analytical programs seek to find closed-form solutions, which means that the

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Analytical programs also include those that, from the analytical solution (i.e., through the Laplace-Transform), seek to find a function that can then be obtained. Often this approach is restricted to linear systems (e.g., a transfer function), and the computation will only proceed correctly under particular preconditions. It is possible to express this through the calculation of eigenvalues.

To b)

Dynamic problems of reactor development are frequently submitted to a conformal computation (i.e., Laplace-Transform) using a digital computer. The numerical Laplace transform can be used as a solution method and yields — in most cases — very large computational effort. The approach is good for those problems of reactor development where the problem can be formulated as a block diagram.

To c)

The statistical methods (also called Monte-Carlo methods) are used only when the problem cannot be solved by a deterministic approach, i.e., is so stochastic in nature that only a statistical approach gives satisfactory results. Such methods are then the appropriate solution method for all random problems.

To d)

The actual dynamic simulation also represents the direct solution to the problem. For this purpose, the simulation is used not only as an aid to computation, but also as an aid for training and for instruction. This is where its great importance lies, and this makes digital simulation a foundation for the processing of dynamic problems.

3.3 Properties of the Systems of Equations

The following aspects should be addressed when considering the properties of the systems of equations:

3.3.1 Analog Equations

All dynamic simulation problems can in principle be treated by partial or ordinary differential equations. In both cases, time always appears as an independent variable. Partial differential equations (e.g., Laplace equations) are generally more complex than ordinary differential equations for the same problem. It is possible to transform them into a set of ordinary differential equations (by spatial discretization, e.g., the finite difference method). The partial differential equation thereby becomes a system of ordinary differential equations.

The direct solution of partial differential equations using an analog computer is achievable in principle. To this end, the analog computer must be used in a particular configuration, however the analog computer achieves this only in a relatively small number of cases. For this reason, the partial differential equation is generally converted into a system of ordinary differential equations, which is then solved numerically. The digital computer offers here the opportunity for integration, and the integration approach is exploited in the form of a simulation language.

3.3.2 Stability Considerations

All dynamic simulation problems can be divided into partial and ordinary differential equations. In both cases, time always appears as the independent variable — at least in the simulation language. For each dynamic simulation, therefore, stability must be investigated. The stability of a system of ordinary differential equations is critical for the following reasons:

The stability of the system of ordinary differential equations must be determined in order to guarantee the correctness of the solution;
The permissible step width and therewith the required computation time depend fundamentally upon the stability properties;
In many practical problems, the system of equations becomes stiff (i.e., the time constants of the system differ by several orders of magnitude). This is particularly the case for reactor-dynamic problems.

For this reason, the stability of the computational procedure is of great importance. The step width control — i.e., the automatic selection of the step width — and integration methods with extended stability must therefore be taken into account for the simulation of reactor-dynamic problems.

3.3.3 Integration Methods

All dynamic simulation problems are solved by step-by-step methods. For this purpose, both partial and ordinary integration can be employed as the solution — it is possible to integrate in the time domain. In both cases, integration with a constant step width and integration with variable step width can be applied.

The step width selection affects the speed of the computation and also the accuracy. Since for reactor dynamics problems both large and very small time constants appear simultaneously — i.e., the system is stiff — the integration step width control must be adaptive (i.e., automatically regulated).

The digital computer offers the advantage here that the integration step width can be selected optimally. A hybrid computer has the advantage of simultaneously integrating on the analog side — and this with the highest possible speed — while the digital side controls the integration step width and the process parameters.

From this it follows that for hybrid computers the possibility exists to exploit the fast integration of the analog part for reactor-dynamic problems and thereby to obtain solutions more rapidly. The step width is determined not by the analog part but by the integration control (digital side). The digital machine therefore retains the necessary flexibility in step width selection, while the analog machine simultaneously retains the advantage of highest integration speed.

3.3.4 Stiff Systems

All dynamic simulation problems can be solved by step-by-step procedures, using either partial or ordinary integration. For stiff systems, ordinary step-by-step integration fails, or at least leads to extremely small step widths. The step width in the numerical integration is limited by the stability criterion. For this reason, there is a need to employ special integration methods for stiff systems — either the implicit integration methods or the hybrid integration techniques.

The digital computer offers here the possibility of using implicit integration methods for stiff systems. In contrast, for analog computation and for hybrid computation, the analog integration part operates unconditionally stably (unconditional stability), which is an advantage. The hybrid computer’s advantage is retained by virtue of the fact that the analog integration part can handle the stiff subsystem while the digital part handles the non-stiff subsystem, using appropriate step-width control.

3.3 Stability of Nonlinear Simulators

The predominantly continuous signals of an analog computer make it somewhat difficult to maintain the stability of the stability-control elements. Many analog computing elements (integrators) have virtually unlimited memory, so that error terms can accumulate. With 100 operations one can hardly still determine whether the system is stable.

Many dynamic simulation problems can be solved by an analog computer directly. Where this is not possible, the much larger state space that can be handled by digital simulation becomes essential. Analog simulation of such systems often requires the use of a stabilizing mechanism.

3.3.5 Stability Problems in Nonlinear Simulators

At a certain point the nonlinearity (of the simulator) becomes noticeably problematic. This concerns the stability (nonlinearity, feedback). Through the use of S&H (sample-and-hold) elements the simulation is stabilized at discrete time steps. This entails stability risks from the digitalization of the variables.

The initialization (re-linearization) creates discontinuities, which can destabilize the system and which carry with them up to S time-step uncertainties.
If the system at S–S time-steps is excited, and has an iterative process controlled alongside it, bet at the step iteration this becomes a positive driver, so at this state the feedback may become positive (increasing system).

This above-mentioned iterative feedback of the problem, as applied through S&H-procedures (3.1.2), leads to the determination of integration variables V and V from (stability). Stability thus prevents the proper integration by an attempt of feedback.

Test example for Re-Oscillation, stronger simulation of the dynamic for convergence condition, quantitative convergence of the Re-Standardization. This is not always achievable by the given steps of initial conditions.

3.3.6 Convergence Condition for Initial Simulator

The Convergence of the Counterpart provides clear initial conditions for the convergence of Begrenzungsfunktionen (3.1.2). The connection is evident from System (3.1.1): Convergence is achieved from Y and V (Stability).

Test example for Re-Stability, stronger simulation of the dynamic for:

Convergence condition for the dynamic simulator (Re-Stability)
Initial condition for the stabilization of the convergence
Stability limits for the stabilization convergence

4. Simulator Types

The comparison of the four simulator types takes place according to the criteria developed in Section 2.1. Here we distinguish between:

a, b, c, d — Qualitative Criteria: Integration, Subtraction, Specification, Boundary, Convergence, Memory, nonlinear functions, logic/trigger functions

4.1.1 Qualitative Comparison of Functions

It is now specified in tabular form the previously mentioned mathematically-related elements of the respective simulators:

CDCP: +, -, /, Quadrature, Integration, Specification, Memory, nonlinear functions, scaling, nonlinear representation of multiple nonlinear functions. This CDCP system is presented and the following accuracy is developed, once the accuracy is developed, each nonlinear function generator generates accuracy.

INB + DAS: +, -, /, Arithmetic Functions, Quadrature, Exponentials, Polynomials, Forecasting (Forward-step), Out. 100 complex Functions. This includes the PDPI32-computing functions.

DIGSYS: Known. Another Function only on the Feasibility scale: from this given program the documentation of system functions.

4.1.2 Programmability

In the simulators in these tables under-consideration exists, when the simulation is performed, digital programs are relatively uncontrolled in size. Any better than DIGSYS has CDCP, which is better than digital in scalability. This CDCP scalability means that a PDPI32 program requires from the size a (Segment), as in the same size DIGSYS is more capable.

For the digital simulation this is especially true when the functionality is well-known.

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The relatively modest accuracy of ISI 7920 makes it hardly usable in step-related control problems. Many dynamic simulators can, however, be well used when they represent several nonlinear conditions (nonlinear, boundary, stabilization).

Many dynamic simulation problems can also be managed by an analog computer, as well. In addition, relatively few dynamic processes can be operated by ISI-Digital ones (relatively few). The Feasibility describes the system as a DIGSYS integration. Also these Stabilization-System functional elements permit the stabilization of dynamic simulation.

3.3.5 Transitional Stability

When a certain problem is initially solved by the nonlinear simulator, then at sufficiently smaller Stages (3.2.3) will be described as follows:

The initialization (re-linearization) creates discontinuities.
If the system at S–S time-steps is excited and has an iterative process alongside it, during the step iteration this becomes a positive driver.
The re-initialization of the system takes a nonlinear step (3.2.3); the process is thereby sometimes a twice-continuous stable stabilization.

This above-described additional iterative reinitialization process is a standard that permits itself, in that its ISI-procedure consists of System stabilizing feedback, in order to obtain those functions, on which the DIGSYS integration is applied.

[page 21: continued — Section 3.5]

3.5 Digital Simulators

For quantitative analysis the following can be specified as equivalent: sufficient programs and boundary-stabilization that also allow direct verification, while the extraction of Integration accuracy (3.2.3) is to be achieved. The Feasibility includes that for the 3.2 boundary problems that are the most significant.

3.5.1 Characteristics of the Transport Operation

The transport is characterized (3.1.3). For the solution of the Integration process (3.1.2): transport is a System of 5 quantities of V and W. The transport-timing uses Stability, Quadrature, Solution, Polynomial, and it contains V and V (stability). Stability prevents the stabilization by an attempt of feedback.

Test example for Transport with Tracking at 15 Stages, 5 Stability Stages:

Minor Feasibility of the Transport
Use of Feasibility in Stage (Ref-Step, Stabilization, Feedback)

3.5.2 Characteristics of a System of Steps with Nonlinear Stability

At twelve steps these Stability Steps are used, giving the Simulator thereby at 5.A the integration (3.1.2): also through the implementation a positive stability becomes achievable when a nonlinear step is solved. Feasibility of a System of System of a continuous step becomes a feedback stability.

Test example for Transport with Stability at 15 Stages:

I Feasibility for Transport
V Feasibility Tracking for Stabilization

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Test example for Transport with Tracking at 15 Stages:

I Feasibility for Transport
V Feasibility for Stabilization of Stages 5.1 (Ref-Step)

3.5.3 Characteristics of a System with Dynamic Stability

The qualitative dynamic stability stabilizes the dynamic System (Test Example 3.1.2). The Simulation of a System with 3.A.1 gives the dynamic integration (3.1.2): also through the implementation a positive stability becomes achievable when a nonlinear step is solved. The Feasibility of a System of a continuous step becomes feedback stability.

Test example for Dynamic Stability with Tracking at 15 Stages:

I Feasibility for Dynamic System
V Feasibility for Tracking Stabilization

3.5.4 Characteristics of a Continuous Dynamic System

The qualitative dynamic stability of a Tracking System (Test example 3.1.2). The Simulation of a System with 3.A.1 is given by the dynamic integration (3.1.2); also from the implementation comes a positive stability, when a nonlinear step is solved.

[page 23: continued]

Test example for Transport with Tracking at 15 Stages:

Feasibility for Transport Stage (Ref-Step, Stabilization)

3.5.5 Characteristics of a Transport System with Digital Simulation

At twelve steps of Digital Simulation the stability steps are used, giving the Simulator at Stage 5.A the integration (3.1.2). Also through the implementation a positive stability is achieved when a nonlinear step is solved. The Feasibility of a continuous step becomes feedback stability.

Test example for the Digital Simulation:

The initialization creates a re-linearization stability.
If the system at S–S time-steps is excited and has an iterative process alongside it, during the step iteration this becomes a positive driver.
The digitally simulated function from (3.1.2), through the integration, is then approximately a twice-continuous stable stabilization.

This above-mentioned iterative feedback of the digitally-simulated problem, which applies the ISI-procedure steps (3.1.2), leads to the determination of stable variables V and V from (stability). Stability thus prevents proper integration by an attempt of positive feedback.

At the digital simulation there is no separate real initial effect. This Stability is indeed achieved, but the digitally simulated function, through the integration, is approximately a twice-continuous stable stabilization. As such the ISI procedure (3.1.2) applies: the digitally simulated step leads to the determination of stable variables, so that the convergence stability becomes
A T ~ 10^(-3) … hence the time-step A-T = q_0 * A_r * T + const. This equation for the Re-Stability of the initial conditions is not yet necessarily assured.

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It is not possible to avoid, if the problem does not allow finer steps. One possibility — which is valid for both the analog and the digital simulation — requires convergence: that no step is taken past the boundary of the stability condition. The Digital Simulation would then be at a certain position (Region of Stability).

Test example for Re-Oscillation, greater simulation of the dynamics with convergence condition, quantitative convergence of Re-Standardization. This is not always achievable from the given steps of initial conditions.

4. Simulator Types

The comparison of the four simulator types takes place according to the criteria developed in Section 2.1. The comparison is made based on those criteria from Section 2.4 as outlined in Section 2.4 longer/shorter/integrator criteria.

4.1 Qualitative Comparison

In this section an overview is given of those qualitative, mathematically calculable functions of the individual simulator types. It is thereby first given for each simulator — for each one of the individual categories:

Subtraction: +, -, /, Quadrature, Integration, Specification, Boundary-conditions, nonlinear functions Logic-trigger functions: Specification, Boundary-conditions, Storage, nonlinear functions, Logic Feasibility functions: Specification, Boundary, Re-Specification, nonlinear functions, Logic

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CDCP: +, -, /, Quadrature, Integration, Specification, Memory (convergence), nonlinear functions, scaling, nonlinear representation of multiple nonlinear functions. This CDCP System presents itself and the following accuracy is developed, so that each nonlinear function generator generates accuracy.

INB + DAS: +, -, /, Arithmetic Functions, Quadrature, Exponentials, Polynomials, Polynomial-step (Forward-step), Out. 100 complex functions. This includes the PDPI32 computing functions.

DIGSYS: Known. Another Function only at the Feasibility scale: from this given program system the documentation of the functions.

4.1.2 Programmability

In the simulators in this table consideration is given to what underpinning exists when the simulation is performed; the digital programs are relatively unconstrained in size. Any item better than DIGSYS includes CDCP, which is better than digital in scalability. This scalability for CDCP means that a PDPI32 program from the size (Segment), as in the same size DIGSYS, is more programmable.

For the digital simulation this is especially true if the functionality is well established.

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The number of available memory locations in a computer-storage represents a limitation. The ratio varies between 100 for ISI, 315 for CDCP, 750 for INB-DAS. An increase of the total memory through multiplication of the integration-memory reduces digital memory capacity to some degree.

One possibility — in this respect analogously to the Digital-Simulation — requires: that the ISI computing program have a Convergence (re-Stabilization) such that a stable memory-state be maintained. The Digital Simulation would be at a position providing better stability.

Test example for the Digital simulation, greater dynamics with convergence condition, greater convergence of re-standardization. These are not always achievable from the given steps of initial conditions.

The following table gives the general structure, the number of storage-elements for the PDPI32-program described above:

	CDCP	INB-DAS	DIGSYS
Number of integrators	100	75	20
Number of Integration-elements for Subtraction	68	75	20
Number of Storage-elements for PDPI32-programs	1 K	1.7 K	50

*) Number of analog-logic-elements counted at 100 times the re-Stabilization capacity.

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4.1.3 Calculation Accuracy

At this point it should be noted that, mfixed, the calculation accuracy of the analog computer is somewhat less than that of the digital simulator, whereas the digital accuracy of the given implementation is relatively high, so that practically all calculations can be made with an acceptable accuracy.

In Example 1 (Transport Problem) the calculation accuracy described in section 3.2.2 can be analyzed. Examples show the systematic error in the interval (3.1.1): the error contains those from Y and V (Stability).

In Example 2 (from Section 3.2.1 it seems clear) that the accuracy is indeed reduced by a factor of 20 from the ISI-digital mode compared with the analog version. The error is in the range of 1 ms (Integration), so that in such problems where accuracy is not the primary consideration a simulation at 20 ms would suffice.

In Example 2 (from Section 3.2.1) the accuracy gives a factor of 20 from the ISI-digital mode. The error is about 1 ms each time; but accuracy is the requirement. In such problems the ISI simulation is to be preferred, because it gives a greater accuracy in the error-step region (approximately 3.2 ms each time).

[page 28: continued — Table]

Problem	Analog CDCP		INB-DAS		DIGSYS
	f [ops/s]	f [ops/s]	f [ops/s]	f [ops/s]	f [ops/s]	f [ops/s]
1a	20 ms ms	f [ops/s] 200 Hz	f [ops/s] 300 Hz	f [ops/s] 300 Hz	f [ops/s] 300 Hz	f [ops/s] 1000 Hz
1b
2		900 ms	130 ms
3	20 ms ms	200 ms	200 ms	not possible
4

( ) = the right accuracy of the re-standardization process

Remark: Feasibility depended on precision in each case on the Feasibility of the computing.

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The solution of Example 4 is still not clear. This Simulation is not yet achievable. Those Simulations of standard digital-analog simulators are however yet underway.

4.1.5 Feasibility

The Feasibility-comparison takes place based on Section 1a and Section 1b criteria (from Section 2.4 longer/shorter factors); the criteria include feasibility for 3 Variables:

Simulator: Analog, CDCP, ISI-DAS, DIGSYS

Simulator	Analog-Capability
16-bit storage	10^(-3)	low
CDCP	—	low
ISI-DAS	7·10^(-3)
DIGSYS	10^(-4)

For CDCP the feasibility with 20 time-steps does not yet become available. For the digital simulation (ISI) the accuracy is therefore 10^(-3) or less, which in its feasibility mode is within the given step range. Already in the feasibility of multiple simulators 2 verifications are made from the feasibility, however, depending on the Tracker cost.

[page 30: continued]

In relation to the cost factor these items should be noted: both the CDCP and DIGSYS can be used as instruments in the ISI 160/85 in computing applications, where the program of the given integration implementation contains it. Both the digital CDCP and DIGSYS bring in the description of the ISI 160/85 program the accurate evaluation of feasibility, and with it the cost of the computational stability; it is practical, in this role, to mention the cost and the re-scaling.

Both digital simulators CDCP and DIGSYS bring in the description of the ISI 160/85 program the accurate cost-performance. It is practical in each instance to give the cost-relation and re-scaling as follows: the ratio cost is accurate ISI 160/85:

Number of summation-factors (step, stability): 1 for CDCP, 2 for DIGSYS
Number of Simulators (step, stability): 1 for CDCP, 1 for DIGSYS
Number of re-Scalings for PDPI32: programs 4 K for CDCP

4.2 Simulator Architecture

Both and CDCP have in their general architectural description a number of computing elements which for simulation purposes suffice — all of them with a PDPI32 program structure that enables the computing accuracy. The ISI computing describes its architecture in terms of the running-cost stability function for programs.

ISI-DAS is used for programs in the running-cost computing architecture. The feasibility for programs is significantly the digital-computing architecture function for programs.

[page 31: continued]

The program-interface is indeed relatively narrow, so that regardless of the same Computers one of them might have multiple Stages. The Analog Computers typically for the Organization needs around 3 items, at most as many as the elements of the computing-architecture allow. The Simulator Controller is therefore considered:

Control-parameter	CDCP	ISI-DAS	DIGSYS
Per Step	3	6	—
Per Calculation	—	—	12
Per Scope	Yes	Yes	12xs

n = number of analog variables

ISI-DAS has the 5 control-parameters for each implementable step-by-step control (e.g., at re-stabilization, parameter, stabilization-function) as described in Section 4.1.

4.2.1 Hierarchical Structure of the Program-Architecture

Of the 3 hierarchical simulators (Analog Computer, CDCP, ISI-DAS), all carry the problem-architecture stability function, all the variables in programming are also capable of it. In the program for the hierarchy, elements at the same architecture are considered, and DIGSYS is also in the same stability. At these functions both of the simulators carry the hierarchy of the Stabilization:
and ISI-DAS is only in the digital feasibility-function of the hierarchy stability function.

[page 32: continued]

and ISI-DAS, which at each Feasibility of the hierarchy is available for the computing-functions of the dynamic stability. Both simulators (ISI-DAS, DIGSYS) have in the same general structure, the same Feasibility functions of the Step-stability.

4.2.2 Feasibility Comparison

In the feasibility tables the program is presented relatively unconstrained, so that regardless of the overall Programs this can be extended — stability program Analog-ISI. These help to provide the overall Programs more available. Stability during programs is especially true with general Analog-ISI feasibility programs. The Feasibility is especially valuable whenever programs are in general available with help through the ISI Analog-stability:

ISI-DAS and DIGSYS are the Feasibility both from the ISI Analog-stability, which means that a certain program form is more available with feasibility.

4.3 Feasibility Comparison

ISI-DAS as well as CDCP have the Feasibility both from ISI Analog-feasibility, which means that certain program structures are more available with specific computing structure. ISI-DAS is therefore particularly for programs in which the ISI computing architecture is a running function of the stability.

[page 33: continued]

4.3.1 Simulator and the Running Computation

It is likewise achievable that, with ISI-DAS, the running computation can be followed, so that both at the same time the Tracking-stability and also the Tracking of the Step (Program) are more easily achieved. For ISI-DAS and DIGSYS the Tracking is each time more achievable from the Program-computing-stability; one can follow with each program ISI and ISI the stability of those programs.

For ISI-DAS and DIGSYS the Tracking is each item with re-stabilization achievable, in that the ISI stability of those programs can be followed — with re-stabilization programs ISI and ISI the stability of those programs.

For the ISI computing: the computing-stability from those programs which can be tracked, so that for each one the tracking and following functions can be maintained, the stability permits. As already noted, the ISI-DAS and DIGSYS are the tracking stability at the computing functions of ISI running programs.

[page 34: continued]

First of all, for the ISI 160/85 computing programs the factor of 0.10 is given as a Verification in the factor 5.10 from the given area. Since the rightmost (best) computing in the ISI 160/85 achieves this computing-step, the tracking is therefore at the 20–25 factor in the ISI-Step.

With the recently introduced Tracking-Scalings of the ISI 160/85, the area of the tracking achievable factor becomes a factor 0.10 increase in the ISI step. The feasibility is, it seems, the ISI-Step. The tracking of the running-computing has the actual factor, which is already the feasibility of the analog step (approximately a factor 3.2 each time) more precise.

By means of ISI 160/85 improvements to the ISI-Step, first of all the ISI 160 with ISI 85 at first from 20 to 25 ms feasibility improvement, the tracking-step of the ISI 160/85 becomes somewhat more reachable. The feasibility of a tracker is somewhat more reachable — that is at each ISI-Step:

ISI has accordingly the tracking precision, which was already reachable with the analog feasibility step (approximately 3.2 ms each time), so that the stability is reachable.
ISI with each step makes tracking more feasible.

[page 35: continued]

Both ISI-DAS and DIGSYS are the tracking programs from the feasibility tracking, which means the specific program structure is available with tracking stability. ISI-DAS is therefore particularly for programs in which ISI computing architecture is a running tracking-function of stability.

For the Tracker-cost comparison the CDCP and ISI-DAS are the same; their ISI computing architecture runs the feasibility of each tracking-function.

5. Digital Comparison of the Running Computation

As a comparison there are indeed more achievable and feasible programs, so that the differences among Simulator types correspond to: in the programs 75–84 the tracking-function remains available; however, for tracking purposes these programs are generally simpler.

From the tracking comparison among ISI-DAS and ISI 160/85 (DIGSYS), the tracking function remains achievable:

ISI has more than 20 tracking programs each
ISI 160/85 tracking more than 20 tracking programs in computing

The tracking functions of ISI-DAS and ISI 160/85 are achievable so that these programs in each feasibility-computation remain achievable, as well as the each ISI-Step. The tracking function for ISI-DAS and ISI 160/85 in the feasibility of ISI-tracking programs can be computed so that ISI and DIGSYS each have more tracking precision of approximately 3.2 ms each time.

[page 36: continued]

Examined are the ISI-DAS-system from a factor of 5 stages, 1.0 order, as well as 1-3 Coefficients into the ISI-DAS stability feasibility. Both the 3 Coefficients for this feasibility are to be noted — for each feasibility the ISI-System is achievable.

The ISI-System is noted at 0.05 and 3 ms stabilization. It is achievable that such programs — as it seems — consist of 3 Coefficients and can be computed so as to make the ISI-System achievable.

Computation System at Stage 5, 1 Stage:

Minor Feasibility for Stability
Use of Feasibility in Tracking (Ref-Step, Tracking, Re-Stability)

For the Tracking System the ISI-DAS and DIGSYS provide the tracking from ISI, which is then the tracking stability. As such the ISI-procedure applies: the feasibility tracking step leads to the computation of stable tracking variables. The convergence stability becomes:

$$S = \left{ 1 + s \cdot (a + b \cdot A) \right} \cdot A - B \tag{1}$$

$$S \text{ (Cost)} = \text{ Factor for the Computation of the Variables}$$

A — Factor for the Computation of the Tracking functions as well as individual variables
B — Factor for the Computation for the Organization of computing
S — Equation of the computing system

Page 37

where

t(Cof) = time for one multiplication (Dichotomus)
t(Add) = time for one addition (Dichotomus)

For the above-described example, one has:

Z = q · r (a + b) + r² (a + b)                    (2)

with:

q = number of linear terms = 25
r = number of coefficients of the linear terms = 3
p = number of non-linear terms = 5
p = average value of computation for one non-linear operation (a.B., b, harmonic) = 5

For the IBM 7074 and 360/65 systems, the following computing times for the elementary operations are obtained:

System	t(Cof)	t(Add)
7074	15·10⁻⁶	700·10⁻⁶
360/65	1.1·10⁻⁶	2.3·10⁻⁶

The factors for real-time calculations can be determined from the above-mentioned quantities:

A = Z
B = Z

For the above-described system of 30 equations, first-order, the computing times for 2 Runge-Kutta steps per second are expected to be:

Runge-Kutta Method / Ordering	7074	360/65
t(Cof)² (computing time)	0.15	0.005
Step accuracy	61.2	7.8

Page 38

From stability considerations, the step size is bounded. The theoretical stable limit of stability for the Runge-Kutta integration step h_max (about 0.02 sec) sets a maximum step size of 0.5·10⁻² at the given limit h = 0.02. The critical step size is therefore around 200 Runge-Kutta steps per second. As this is already 200 steps per second, a digital computer is needed that has about 200 computing cycles per second. Only the IBM 360/65 satisfies the latter requirement.

The Runge-Kutta method represents an implicit numerical integration method. The computing time t can be estimated as:

t = 2 a (1a + 6b) ≈ a · b                          (3)

Below, for the Runge-Kutta method the values of A, B, c, and the average step sizes are shown, as calculated:

Method	A	B (step accuracy)
…	0.29	0.008
	27.8	3.8

Due to the step sizes, the values of A and B (the size of the average step) are very similar; the resulting relative accuracy (as shown in formula [3, p. 1213]) is checked by the factor. This is therefore due to the favorable stability of integration.

A reduction of step size in the software is easily possible and leads to a simulation in which the simulator can program real-time calculations.

Page 39

— 35 —

For sufficiently complicated problems (such as a.B., the variables cannot be held quite stable by integration methods), one can show that a simulation in real time is largely not feasible for a digital computer. All such attempts, however, lead step by step to ever-increasing degradation of computing time. Importantly, the fully digital-computer-controlled simulation cannot be executed faster than real time.

These elevated real-time constraints arise always — and not only for the digital computer simulator itself — but also for the reasons that integration steps, if too large, lead to unstable numerical results. It follows from this that the requirements for a real-time analog simulation are in the range from factor 10 to 100 faster than for digital methods.

This is in connection with a Runge-Kutta method or implicit integration that the variable steps do not always proceed. Due to the favorable integration, the digital-computer methods can, in principle, reach suitable performance.

4. Summary

The comparison for four hybrid simulators:

Hybrid Simulator	DISA	(for IBM 3250/3000)
DISP	(for IBM 11500)
PDSP	(for IBM 360)

with typical simulation computing time and the listed numerical methods, leads to significantly reduced operating requirements compared to purely digital approaches.

The digital computer DISP includes (by weight) full language power as its list of integration constants, the problem of the machine; it is here a significant additional advantage compared to other machines in the numerical calculation, and partly in the factor of 10 for problems. It can thereby simulate larger systems on the factor 5 to 10 fewer.

Page 40

— 36 —

A certain advantage over the other digital computer is achieved here, whereby it exhibits the advantage that the programs can be entered in the solving step somewhat more easily and with fewer steps.

For the analog computer, all problems can be run minimally in a few steps. Digital computers of this type show a computation time of about 1000-millisecond order. It is noted that the requirements of the digital computer are clearly favorable and that the machine can run a factor 3 to 10 faster than a simulation program. The machine is limited by a factor 5 to 10 only for problems of the digital calculations.

For the preparation of systems on the digital computer, the computation time is clearly shorter when the following requirement is noted: it must be so noted that the digital computers of the IBM 360 can use a factor 5 to 10 faster than a Simulator. In this step, a factor 3–10 faster is not always possible.

The preparation of programs from IBM 360-DAS or DIISB may need to be taken into account for longer computation time for real-time calculations, so that the computation times of a single step are about 3–5 shorter, noting the preparation time.

Page 41

— 37 —

Digital simulators such as DIISB and ISB-DAS are also applicable for special digital simulators that can simulate more problems on the factor 5 to 10 and faster. The problem is clearly dependent on the factor 5 to 10 in the simulation of large systems.

The application of digital simulators in the context of Reactor Safety Investigation serves the following specific purposes:

Development of a further digital simulator (DIISB based on IBM 11500 and IBM 360) as an integration computing system; it is intended for the simulation of large linear differential equation systems. As a Simulator, it is suitable for use in large long-term calculation tasks and real-time simulations requiring computing performance.
Development of a further Digital Simulator for the integration, ISB-DAS and DIISB, as a computing device for real-time use, requiring special digital computing power. The integration programs from IBM-DAS or DIISB can be structured for long-term calculation tasks.
Preparation of programs from the IBM 360 or DIISB being added to the time frame for target-accurate computing; this requires that the simulator in computer mode has access to the additional real-time simulation program (computing completion time).

Page 42

— 38 —

The programming of the analog computers requires a relatively short computing time and is for many simple operations a clearly manageable function and programming task.
The real-time requirements, especially for the analog computer, are better met in the sense that the computing time is further reduced. Hybrid computers can perform even for certain computing requirements a factor of 5 to 10 faster than digital computers; they allow relatively easy computing tasks to be accomplished and can also process additional larger computing programs in digital computing mode.
Hybrid computers, which can display all problem functions with analog computing, are well suited for the real-time presentation of problem-solving tasks: from the analog mode it is easy to extend and expand the computing program; those tasks which especially require very large operating computing loads can be included here.

From the analog computing mode, digital data in the larger amount can now be added with an associated digital analog interface; it is a requirement that the hybrid computer programming also accepts the required computing analog program. Thus the required digital data is displayed for the computing program.

The digital programming of Hybrid simulators should involve a separate framework for calculating the Analog program alongside digital computation; this will mainly encompass the digital computing programs together with integration of the Hybrid computer.

The programming examples that are described in the text originate from the work of B. Janich, K. Kath, and B. Sarton distributed here.

Page 43 — Bibliography

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Transient Temperature Calculations Using a Hybrid Computer
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[7] Boudinot C.K., et al.
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International Symposium on Analog and Hybrid Computation Applied to Nuclear Energy, Versailles 1968

Page 44

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Analog-Rechenprogramm für kurzfristige schnelle Reaktoren mit Sicherheitsabstand [Analog computing program for short-term fast reactors with safety margin]
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[10] Calverola L., et al.
The Balanced Oscillator Todc in SOPHIR
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Automatic Control Systems for Balanced Oscillator Tasks
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Hybrid simulation of Monte Stribkov for ruling test transfer policies on initial boundary
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[13] Reuther R.J.
Monte Carlo Solution of partial Differential Equations using a Hybrid Computer
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Rechenprogramm für die Berechnung der Eigenprogramme der Neutronenfluss mit Hilfe des Schutzpotential-Optimierungsverfahren [Computing program for the calculation of eigenvalues of the neutron flux using the protective-potential optimization method]
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[15] Calverola L., Ballenchieschi G.O.
Transient Temperature Solutions with Spatial Variables
First Parts Solid Analysis
KFK 271, Vol 1964

Page 45

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Zur numerischen Lösung der ströhungsbedingten dynamischen Gleichungen eines schnellen Leistungsreaktors mit Hilfe des Wechselwirkungsprinzips [On the numerical solution of flow-induced dynamic equations of a fast power reactor using the principle of interaction]
KFK 391/I, Sept. 1965

[17] Fischer M.
Zur Lösung von Vollzustands-Differentialgleichungen unter Berücksichtigung von Sicherheitsabstandsbedingungen [On the solution of full-state differential equations with consideration of safety-margin conditions]
Kernforschungszentrum Karlsruhe, E/52/A, Sept. 1967

[18] Babula, Dethier, and Batermach M.
In Reactor Core with Diplomas with the Dynamic Simulation in Fast Breeder Reactors and Hybrid Computing Systems
KFK 799, December 1968

[19] Sohn P.P.
Time sharing real-time analog equipment
Simulation, Febr. 1968

[20] Vichnevetsky R.
Analog/Digital Solution of Partial Differential Equations in the Nuclear Industry
International Symposium on Analog and Hybrid Computation Applied to Nuclear Energy, Versailles 1968

[21] de Bartolomeis M., et al.
Improving the analog simulation of partial differential equations by hybrid computation
Simulation August 1968

Page 46

[22] Rosenblum R.J.
On the Hybrid Solution of Partial Differential Equations of the Hyperbolic Type
International Symposium on Analog and Hybrid Computation Applied to Nuclear Energy, Versailles Sept. 1968

[23] Ansler A.J.
A new stable computing method for the serial hybrid computer integration of partial differential equations
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Die Lösung von bis zu 1000 PDE Leitungsgleichungen schneller Reaktionsdynamik [The solution of up to 1000 PDE conduction equations of fast reaction dynamics]
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DIISA – Digital-Computerprogramm zur Koeffizientenberechnung, Potenzflussoptimierung und Prüfung von Analogprogrammen [DIISA – digital computer program for coefficient calculation, power-flux optimization, and verification of analog programs] (not yet available)
KFK-Conf. 1967, Lausanne

[26] Frisch V.
DIISB–DIISA-Computerprogramm zur Koeffizientenberechnung, Potenzflussoptimierung und Prüfung von Analogprogrammen [DIISB–DIISA computer program for coefficient calculation, power-flux optimization, and verification of analog programs] (not yet available)

[27] APCORE: Analog Programming and Checkings
Programmers Manual
IBM 1967 ×

[28] Peterson T.H.
MIDAS – How it works and how it is worked
Fall Joint Computer Conference 1964
Spartan Books Inc., Baltimore, Md.

[29] Peterson T.H. and Lugar C.K.
”PROTOLIO” – A digital analog simulator program for the IBM 1620
Fall Joint Computer Conference 1964

Page 47 — Bibliography continued

[31] IBM Application Program
1130 Continuous System Modeling Program (1130-CX-13X)
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[32] IBM Application Program
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[33] Hurley J.R. and Skiles J.J.:
DYSAC – A digitally simulated analog computer
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[34] Trauboth H.:
Programmsystem zur Simulierung allgemeiner Regelsysteme auf dem Digitalrechner [Program system for the simulation of general control systems on the digital computer]
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[35] Syn W.M., Linebarger R.N.:
DSL/90 – A digital simulation program for continuous system modeling
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[36] The SCI Continuous System Simulation Language (CSSL)
Simulation, Vol 9, Nr. 6, Dez. 1967

[37] Harnett R.T.:
MIDAS – An Analog Approach to Digital Computation
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Page 48 — Bibliography continued

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Electronic Analog and Hybrid Computers
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A medium-scale hybrid interface
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[40] Korn G.A.:
Computer Interface Design for Simulation and Instrumentation in Nuclear Engineering
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Applying Hybrid Computers
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Hybrid-Rechenanlagen [Hybrid computers]
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Page 49 — Bibliography continued

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Second generation language orientated hybrid computers
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Dynamische Simulatoren in der Reaktorentwicklung. Ein Vergleich

Nuclear Research Center Karlsruhe

Table of Contents

1. Introduction

2. Dynamic Simulators

2.1 Criteria for Evaluating Simulators

2.3 Development Trends in Simulators

2.3.1 Hybrid Simulators

2.3.2 Digital Simulators

2.3.3 Selected Simulators

2.4.1 TRICE — EAI

2.4.2 IBM 360/75S

3. Problem Analysis

3.1 Dynamic Problems in Reactor Development

3.2 Solution Methods

3.3 Properties of the Systems of Equations

3.3.1 Analog Equations

3.3.2 Stability Considerations

3.3.3 Integration Methods

3.3.4 Stiff Systems

3.3 Stability of Nonlinear Simulators

3.3.5 Stability Problems in Nonlinear Simulators

3.3.6 Convergence Condition for Initial Simulator

4. Simulator Types

4.1.1 Qualitative Comparison of Functions

4.1.2 Programmability

[page 20: continued]

3.3.5 Transitional Stability

[page 21: continued — Section 3.5]

3.5 Digital Simulators

3.5.1 Characteristics of the Transport Operation

3.5.2 Characteristics of a System of Steps with Nonlinear Stability

[page 22: continued]

3.5.3 Characteristics of a System with Dynamic Stability

3.5.4 Characteristics of a Continuous Dynamic System

[page 23: continued]

3.5.5 Characteristics of a Transport System with Digital Simulation

[page 24: continued]

4. Simulator Types

4.1 Qualitative Comparison

[page 25: continued]

4.1.2 Programmability

[page 26: continued]

[page 27: continued]

4.1.3 Calculation Accuracy

[page 28: continued — Table]

[page 29: continued]

4.1.5 Feasibility

[page 30: continued]

4.2 Simulator Architecture

[page 31: continued]

4.2.1 Hierarchical Structure of the Program-Architecture

[page 32: continued]

4.2.2 Feasibility Comparison

4.3 Feasibility Comparison

[page 33: continued]

4.3.1 Simulator and the Running Computation

[page 34: continued]

[page 35: continued]

5. Digital Comparison of the Running Computation

[page 36: continued]

Page 37

Page 38

Page 39

4. Summary

Page 40

Page 41

Page 42

Page 43 — Bibliography

Literature

Page 44

Page 45

Page 46

Page 47 — Bibliography continued

Page 48 — Bibliography continued

Page 49 — Bibliography continued

Page 50 — Bibliography continued

Page 51 — Bibliography continued