Master for Co-Simulation Using FMI

Transcription

1 Master for Co-Simlation Using FMI Jens Bastian Christoph Claß Ssann Wolf Peter Schneider Franhofer Institte for Integrated Circits IIS / Design Atomation Division EAS Zenerstraße 38, 69 Dresden, Germany {Jens.Bastian, Christoph.Class, Ssann.Wolf, Peter.Schneider}@eas.iis.franhofer.de Abstract Co-Simlation is a general approach to simlate copled technical systems. In a master-slave concept the slaves simlate sb-problems whereas the master is responsible for both coordinating the overall simlation as well as transferring data. To nify the interface between master and slave the FMI for Co- Simlation was developed. Using FMI a master was implemented with simple and advanced algorithms which can be applied depending on the properties of the involved slave simlators. The master was tested amongst others by copling with SimlationX. Keywords: co-simlation; FMI; master Introdction Modeling problems in natral sciences and engineering often leads to hybrid systems of differential and algebraic, time continos and time or event discrete eqations. Often complex mlti-disciplinary systems cannot be modeled and simlated in one simlation tool alone or sbsystem models are available only for a specific simlation tool. Sometimes sb-problems shall be simlated with the simlator which sits best for the specific domain. Ths for the simlation of mlti-disciplinary problems or for hardware-in-theloop simlation it is often reasonable or even necessary to cople different simlation tools with each other or with real world system components. Simlator copling is sed in varios fields of application like atomotive engineering, microelectronics, mechatronics etc. Up to now simlator copling is nearly always a point-to-point soltion tailored to the involved simlators. These special soltions case high effort so a generally accepted interface for simlator copling spported by many simlation tools is desirable. Co-Simlation Co-simlation is an approach for the joint simlation of models developed with different tools (tool copling) where each tool treats one part of a modlar copled problem. Intermediate reslts (variables, stats information) are exchanged between these tools dring simlation where data exchange is restricted to discrete commnication points. Between these commnication points the sbsystems are solved independently.. Copling of simlators A simlation tool S can be copled if it is able to commnicate data dring simlation at certain time points t, cf. Figre. Here inpt variables are denoted by and otpt variables by y. Figre : Block representation of a simlator S Simlators have different capabilities which have an inflence on the algorithms that can be sed for their copling. Sch capabilities are: The simlator can handle variable commnication step sizes. The simlator can handle events. It is possible to ndo a time step, i.e. the simlator can reject time steps. When sing simlator copling the original problem is divided into N sbproblems each handled by a simlator. Typically, N is small, i.e. below. Thereby the simlators do not have to be different. 5

2 The signal flow for the copled simlators can be described by a directed graph with the simlators as the nodes and the exchanged data as the edges. If there is feedback in the graph then cycles exist. A cycle is a path in a graph with the same node as start and end point. Cycles can be eliminated if the simlators in a cycle are combined into a spersimlator i.e. a simlator sperior to the simlators of the cycle. Figre shows an example of sch a graph. Simlator A has the highest priority. The simlators B, C, and D form a cycle. E, F, and G are sbordinated to this cycle. That means simlator A is exected first of all. Then the cycle of B, C, and D is finished. Afterward simlators E and F are exected whereat both simlators can be rn in parallel. At last G is processed. Figre : Example graph of copled simlation tools For simlation, the whole graph is analyzed first. If cycles are detected then they are combined into a sper simlator. The simlators are copled with directed data flow. A priority is assigned to each simlator with representing the highest priority. Simlators with the same priority can be exected in parallel. All simlators in cycles either have to be processed iteratively or with small enogh time steps and error control. synchronizes, controlles and manages them []. Each edge of the graph is regarded as to go throgh the master, cf. Figre 3. The master serves as an interface, establishes connections and exchanges data between the simlators which are called slaves. Slaves are assmed to commnicate with the master only.. Basic Co-simlation comptational flow The whole co-simlation can be divided into several phases.. Initialization phase All simlation tools are prepared for starting the cosimlation. The master receives the properties of the slaves. Frthermore the master receives the connection graph. The slaves and models are initialized and parameters are set. The commnication links between master and slaves are established. The master chooses its algorithm based on the capabilities of the slaves as well as the connection graph and ser inpt.. Simlation phase The master forces the slaves to simlate the time interval from start time to stop time by stepwise solving master sbintervals which are also called commnication steps. Their bondaries are called commnication points. In case of event iteration the commnication step size can be zero. The simlation is performed independently for all sbsystems restricting data exchange between sbsystems to these commnication points. Before simlating a sbinterval a slave receives its inpt vales and possibly their derivatives with respect to time as well as the commnication step size from the master. After finishing the commnication step the master receives the otpt vales of the slave and possibly their derivatives with respect to time. Frthermore the slave stats has to be transferred to the master. If the slave simlation fails frther commnication is necessary. 3. Closing phase The master stops the complete simlation and is responsible for proper memory deallocation, terminating and resetting or shtting down the slaves..3 Accracy and stability Figre 3: Master-Slave strctre Instead of direct copling, a master is assmed to be located between the single simlation tools which Co-simlation can lead to problems regarding stability and accracy of the simlation [] especially if feedback exists between simlators, cf. the example given in section 4.4. If a simlation tools provides an 6

3 interface for co-simlation at all then sally it is not possible to reset a simlator so that a time step can be repeated e.g. with a smaller step size. So co-simlation shold often be sed as a last resort as long as iterative methods have to be sed for stability and simlators only provide a rdimental co-simlation interface. Hopeflly this will change in ftre with the introdction of a standardized cosimlation interface like the one proposed in the next section. 3 Fnctional Mock-p Interface (FMI) for Co-Simlation The Fnctional Mock-p Interface (FMI) for Co- Simlation [3], [4], [5] is an interface standard for the soltion of time dependent copled systems consisting of sbsystems that are continos in time or time-discrete. It provides interfaces between master and slaves and addresses both data exchange and algorithmic isses. Both simple as well as more sophisticated master algorithms are spported. However, the master algorithm itself is not part of FMI for Co-Simlation. FMI for Co-Simlation consists of two parts: Co-Simlation Interface: a set of C fnctions for controlling the slaves and for data exchange of inpt and otpt vales as well as stats information. Co-Simlation Description Schema: defines the strctre and content of an XML file. This slave specific XML file contains static information abot the model (inpt and otpt variables, parameters, ) and the solver/simlator (capabilities, ). The complete interface description can be obtained from [3]. The capability flags in the XML file characterize the ability of the slave to spport advanced master algorithms which se variable commnication step sizes, higher order signal extrapolation etc. A component implementing the FMI is called Fnctional Mock-p Unit (FMU). It consists of one zip file containing the XML description file and the implementation in sorce or binary form (dynamic library). A master can import an FMU by first reading the model description XML file contained in the zip file. Copling simlators by FMI for Co-Simlation hides their implementation details and ths can protect intellectal property. FMI for Co-Simlation version. was pblished in October. Crrently it is planned to combine FMI for Co-Simlation with FMI for Model Exchange to an FMI standard. 4 EAS Master MODELISAR [6] is a research project within the Eropean ITEA program. It is aimed to develop the FMI as well as to spport it by involved tool vendors. Use cases will show the benefits of applied FMI. Master algorithms are not standardized with FMI bt developed in the MODELISAR project e.g. by tool vendors. A prototypical implementation of a master has been provided by EAS for the MODELI- SAR consortim. The package contains the ANSI C code of the master, a generic C fnction slave, and a collection of examples. The C fnction slave provides the basic fnctionality of FMI for Co-Simlation. The ser has only to provide two fnctions for initialization (the nmber of inpt and otpt variables) and the comptation of a step with the step size commnicated by the master. 4. Configration The master is configred by a simple text file. There are keywords for start and stop time, step size, copling algorithm, error tolerance etc. The copled FMUs with their paths have to appear within the configration file, too. The graph of the simlator copling has to be spplied by an incidence matrix and information abot the priority of the slaves as well as occrring cycles. 4. Copling algorithms The master prototype provides three algorithms for the simlation with fixed step size: data flow between the slaves withot iterations, i.e. simple forward calclation fixed point iteration of all cycles within the graph simple implementation of Newton s method with Jacobians approximated by finite differences All master algorithms proceed in macro steps of fixed step size from start time to end time. The comptation of a time step from t i to t i+ within cycles is performed in the following way: Every slave makes an assmption for its inpt vale at time t i+. Crrently this is done sing constant t, i.e. in each macro step interpolation i t i 7

4 all terms that cople the sbsystems are frozen. Ths synchronization and pdate of the exchanged vales with compted otpt y t i is done at the end of the time step. Becase no slave depends on the crrent otpt of another one, the slaves can rn in parallel. This iteration scheme is called to be of Jacobi type. Another approach wold be to simlate a time step with every slave of a cycle one after another and to se the otpt y t i jst calclated as inpt t i for the following slaves. These staggered algorithms which handle the sbsystems seqentially are called of Gaß-Seidel type. This method was sed within a first master implementation. The drawback of this approach is that the slaves within the cycles cannot rn in parallel and the behavior of the iteration depends on the calling seqence of the slaves. However, an example exists where this approach converges while the first method does not converge. 4.3 Simple slave test examples A collection of examples sing the C fnction slave is provided together with the master. They cover different types of copling with or withot cycles, nonlinear eqations, ODEs, DAEs and demonstrate the sage of the configration file. Some of the examples can be solved with all master algorithms, some only with Newton s method. One of these examples is BspK6. It consists of for copled slaves which exchange 4 vales (,,, 3) of type fmireal and vales (4, 5) of type fmiinteger, cf. Figre 4. The slaves S, S, and S form a cycle. S y y S Slave S: Slave S: Slave S3: y : x : x 4 3 d x x sin3πt d t y x sin πt d x x sinπt d t y x sin πt y sin πt sin πt Figre 5 and Figre 6 show simlation reslts for constant step size -4 and Newton s method as iterative method for the cycle. The other two methods do not converge for this example BspK t Figre 5: Simlation reslts for exchanged vales and y 3 S BspK6 4 5 y.6 S3.4 Figre 4: Example BspK6 from collection Inpt, otpt, and internal variables of the slaves are related by the following eqations. Slave S: d x x d t t Figre 6: Simlation reslts for exchanged vales,, 4, and 5 8

5 4.4 Copling with ITI SimlationX Another example shows copling of SimlationX [7] with a C fnction slave via the EAS Master. The original SimlationX model is shown in Figre 7. It is a simple plant with a controller driven by the speed fnction rpm. s t s f t As it can be seen, the anglar velocity of the plant shows a small bt fast decaying oscillation in the original model after the speed has been switched to after s. In contrast, the oscillation is larger and does not decay in the simlator copling. For larger step sizes the amplitde of this oscillation is even larger (not shown). At the moment, SimlationX cannot discard steps so a simlation with iterative methods was not possible. With iterative methods we expect the oscillation to decay like in the original model. 4.5 Efficiency Figre 7: Fll SimlationX model This model has been split into three FMUs: two SimlationX FMUs for the controller and the plant and one C fnction simlator FMU for the speed inpt, cf. Figre 8. The SimlationX FMUs contain the model as well as the solver as a DLL. They were created via the code export option of SimlationX. Efficiency and simlation speed strongly depend on the problem which has to be solved. Clearly, the most efficient approach wold be to se only one simlation tool and do withot cosimlation. If this is not possible then problems described by graphs withot feedback can be simlated most efficiently sing the non-iterative method. If there are cycles within the graph and no iterative methods can be sed becase the simlators cannot discard steps then accracy and nmerical stability may be poor. Anyway, the macro step size has to be very small then and ths the comptational costs strongly increase. Controller Speed S S Plant Figre 8: Copling of three FMUs Plant.sensor.om (rad/s) SimlationX Copling.5.5 time (s) Figre 9: Simlation reslts The copled FMUs have been simlated by the prototypical master with fixed step size -3 with the simple algorithm for forward calclation withot iteration. Reslts of this calclation as well as of the original simlation model are presented in Figre 9. S3 Figre : Disadvantage of crrent OpenMP approach compared to thread programming By sing OpenMP [8] slaves of the same priority can rn in parallel. However, the crrent implementation of this approach has a disadvantage compared to thread programming which will be explained with the help of Figre. Here S has a higher priority than S. S3 can have the same priority either as S or S. Ths either both S3 and S can rn in parallel and the simlation contines with S after both S and S3 have finished or S and S3 can both rn in parallel after S has finished. Instead it wold be better to handle S and S as a sper slave which rns in parallel with S3, i.e. synchronization takes place at the start of S and S3 and after S and S3 have finished. However, either a more complicated data strctre has to be sed for this prpose if OpenMP shold be sed or platform dependent thread programming has to be sed. 9

6 4.6 Smmary of properties The implementation of the EAS Master is as platform independent as possible. Platform dependent code mainly for dealing with dynamic libraries cold happily be collected as preprocessor defines within a single header file. Ths the master rns on mltiple platforms (MS Windows, Linx, Sn Solaris). Slaves can rn in parallel if they have the same priority. Platform independence also was the reason to se OpenMP instead of explicitly dealing with thread programming for this prpose. OpenMP is spported by newer version of the major C compilers (gcc, Visal Stdio). Parallelization is realized by one #pragma directive in front of a for loop so that compilers withot OpenMP spport simply compile the code for serial exection. However, the OpenMP approach has the drawback compared to explicit dealing with threads that only slaves of the same priority and not across different priorities can rn in parallel. Crrently the three algorithms mentioned in section 4. are available. 4.7 Ftre enhancements A commercially available version of the master will have the following featres: The graph will atomatically be analyzed for the priority of the slaves and cycles. Newton s method will be improved. A better Jacobian pdate strategy will be sed so that the high cost of calclating a new Jacobian by finite differences will be redced. Broyden s method will be available as another iterative method. A step size control will be implemented based on reslts in [9] so that variable macro steps can be sed. Polynomial interpolation of data besides the crrently sed constant interpolation will be spported. 5 Conclsions Co-simlation is a powerfl method to simlate heterogeneos systems where each sbsystem is simlated by its own specialized simlator. However, crrently simlation tools have their own interface for copling if at all. Additionally, they are often not able to discard steps and ths not sitable for iterative methods. The Fnctional Mock-p Interface (FMI) for Co- Simlation as a proposed standard for simlator copling will hopeflly be widely sed becase it replaces crrent point-to-point soltions and ths eases the rese of models tailored to special simlators. The protection of intellectal property is also possible with FMI. Providing the prototypical master implementation will hopeflly help to promote the FMI for Cosimlation. Acknowledgements The SimlationX model and FMUs were kindly provided by T. Blochwitz from ITI. This work is spported by the German BMBF within the ITEA MODELISAR project. The athors thank the reviewers for valable remarks. References [] Wolf, S.; Blochwitz, T.: Master Slave Simlator Copling. ITI Symposim. [] Schierz, T.; Arnold, M.: Advanced nmerical methods for co-simlation algorithms in vehicle system dynamics. st Conference on Mltiphysics Simlation, Bonn. [3] Fnctional Mock-p Interface for Co- Simlation v., MODELISAR consortim,. [4] Arnold, M.; Blochwitz, T.; Claß, C.; Neidhold, T.; Schierz, T.; Wolf, S.: FMI-for- CoSimlation. st Conference on Mltiphysics Simlation, Bonn,. [5] Enge-Rosenblatt, O., Claß, C.; Schneider, A.; Schneider, P.: Fnctional Digital Mockp and the Fnctional Mock-p Interface Two Complementary Approaches for a Comprehensive Investigation of Heterogeneos Systems. 8 th International Modelica Conference, Dresden,. [6] [7] [8] [9] Schierz, T.; Arnold, M.; Eichberger, A.; Friedrich, M.: Stdy on Theoretical and Practical Aspects of Commnication Stepsize Control. MODELISAR, swp3 report,.