Parallel Multiple-Shooting and Collocation Optimization with OpenModelica

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1 arallel Multiple-Shooting and Collocation Optimization with OpenModelica Bernhard Bachmann 3, Lennart Ochel 3, Vitalij Ruge 3, Mahder Gebremedhin 1, eter Fritzson 1, Vaheed Nezhadali, Lars Eriksson, Martin Sivertsson 1 ELAB rogramming Environment Lab, Dept. Computer Science Linköping University, SE Linköping, Sweden Vehicular Systems, Dept. Electrical Engineering Linköping University, SE Linköping, Sweden 3 Dept. Mathematics and Engineering, University of Applied Sciences,D Bielefeld, Germany {bernhard.bachmann,lennart.ochel,vitalij.ruge}@fh-bielefeld.de, {peter.fritzson,mahder.gebremedhin,vaheed.nezhadali,marsi}@liu.se Abstract Nonlinear model predictive control (NMC) has become increasingly important for today s control engineers during the last decade. In order to apply NMC a nonlinear optimal control problem (NOC) must be solved which needs a high computational effort. Stateof-the-art solution algorithms are based on multiple shooting and/or collocation algorithms, which are needed to solve the underlying dynamic model description. This paper concentrates on parallelizing these time-consuming algorithms, which finally lead to a very fast solution of the underlying NOC suitable for on-line application. The modeling and problem description is done in Optimica and Modelica. The simulation is performed using OpenModelica. Speedup curves for parallel execution are presented. To our knowledge this is one of the first parallel multiple shooting and collocation algorithms with corresponding implementations. Keywords: Modelica, Optimica, optimization, multiple shooting, collocation, parallel, simulation 1 Introduction This paper presents efficient parallel implementations and measurement results of solution methods for nonlinear optimal control problems (NOC) relevant for nonlinear model predictive control (NMC) applications. NMC as well as NOC have become increasingly important for today s control engineers during the last decade [3], []. State-of-the-art solution algorithms are based on multiple shooting and/or collocation algorithms [], which are needed to solve the underlying dynamic model description. This paper concentrates on parallelizing these time-consuming algorithms, which finally lead to a very fast solution of the underlying NOC suitable for on-line application. The modeling and problem description is done in Modelica extended with optimization goal functions and constraints specified as in Optimica [18]. The simulation is performed using OpenModelica. Speedup curves for parallel execution are presented. To our knowledge this is one of the first parallel multiple shooting and collocation algorithms with corresponding implementations. Section describes the underlying mathematical problem formulation including the objective function and constraints to the state and control variables. The general discretization scheme applied is discussed in Section 3. This general approach gives the possibility to apply multiple shooting as well as collocation algorithms to the problem, which is described in Section. Different applications, some of them industrial driven, have been investigated and are described in Section 5. arallel execution of the constraint equations of the NOC is performed in Section 0. The results show reasonable speedups of the optimization time when it comes to time consuming calculation of the model equations. The necessary implementations are partly realized in the OpenModelica Compiler, which is described in Section 7. The paper concludes with a summary of the achieved results. The Nonlinear Optimal Control roblem (NOC) The numerical solution of NOC is performed by solving the following problem formulation [8][9]:

2 ,,,, (.1) subject to,,,, 0 " 0 (.) (.3) (.) (.5) where #$ % & and #$ % * are the state and control variables, respectively. The receding time horizon is given by the interval +,,. The constraints (.), (.3), (.) and (.5) describe the initial conditions, the nonlinear dynamic model description based on differential algebraic equations (DAEs, Modelica), the path constraints,,#$ - and the terminal constraints. Support for time-optimal control and corresponding terminal constraints is work-in-progress and are not yet provided by the current implementation..1 Boundary Value roblems The objective function (.1), that needs to be minimized, includes conditions at the boundary time point t / stated by the function as well as conditions taking into account the whole time horizon stated by 0,,. The function describes conditions that should be fulfilled at the final time point similar to the terminal constraint (.5). Since is part of the objective function,, the applied optimization methods may not find a solution that fulfills the corresponding terminal constraints, but is very close to it. The trajectories are influenced by changing the control variables. Different trajectories using different control variables are visualized in Figure 1. Figure 1. Different trajectories achieved by varying control variables. Only one trajectory fulfills the terminal constraint (red dot). [17] On the other hand, different trajectories could fulfill the same terminal constraints. Taking into account the whole time horizon by minimizing the second part 0,, of the objective function will lead to the selection of the optimal trajectory. This behavior is visualized in Figure. Figure. Different trajectories that fulfill the terminal constraint. [17] 3 General Discretization Scheme In order to apply a general discretization scheme the NOC formulation is rewritten to a general form which later can be used to derive the different possible numerical algorithms e.g. multiple shooting, the collocation algorithm, etc. [7]. Equation (.3) can be rewritten as follows:,, dt. (3.1) When discretizing the time horizon +,, into a finite number of intervals 3, 5,,3 78, 7 5 (e.g. equidistant partitioning: 9 : ;<=,=0,>,, ;? 8 7 ) equation (3.1) can be further reformulated to

3 ,, dt D8 dt. EF A Each integral on a subinterval ABC,, dt A (3.) can now be treated independently, if additional constraints are added to the NOC formulation to force the calculation of an overall continuous solution. Therefore, locally the problem reduces to a boundary value problem [5] stated by IBC G GH G G,, (3.3) I for 0,>,J1. It yields G G G. Continuity is forced by additional constraints G GH GH, which finally leads to a boundary value problem. Each sub-problem (3.3) can be solved independently and in parallel. By varying the control variable in each sub-interval the solution of (3.3) can be influenced in order to fulfill the overall continuity constraints. In the current approach it is assumed that G is constant for each subinterval 3 G, GH 5. NOC Discretization Scheme Numerical Treatment of Integration Figure 3. Schematic view of the algorithmic dependencies. Alternative Numerical Treatment of Integration.1 NOC Applying Shooting Different numerical methods are available to solve equation (3.3). The first approach presented within this paper is the reformulation of (3.3) to an ordinary differential equation G G, E, (.1) with the initial condition G G G. In order to solve equation (.1) the solution can be achieved by choosing an appropriate integration algorithm. (e.g. methods suitable for stiff differential systems) already available in OpenModelica.. NOC Applying Collocation The second approach involves the well-known collocation methods, which are summarized in the following section. Basically, equation (3.3) is approximated by a quadrature formula based on the following considerations. First, a substitution is defined to transform the integral boundaries of formula (3.3) to the standard interval [0,1]. IBC G,, I ;< G L G M, G,L G M M L G M G ;<M This derived integral can now be approximated using polynomial approximations for the states []: G G L G MN@O 9 M 9F GL G MN@O9M 9F < 9,G < 9,G (.) (.3) Alternatively, the polynomial approximation can be used for the derivatives of the states [10] GL G MN@O 9 M 9F <Q 9,G GL G MNR O 9 SS<Q 9,G 9F whereby the polynomial approximation is described using the Lagrangian polynomial: T (.3) (.)

4 O 9 M U MJM V M 9 JM V VF VW9 (.5) The first approximation is less flexible than the second, since fewer free parameters are given (i.e., the integration constant in (.5) ), which finally leads to a better approximation of equation (3.3) when using the second approach. On the other hand the first approximation leads to a simpler equation system which is easier to solve within each interval. Taking into account the above described transformation of each discretization interval to the interval [0,1] further reformulations are necessary. The states are redefined by: GL G MNXY G L G MX G M GL G MNZY G L G MZ G M (.6) (.7) The basis function must fulfill the boundary conditions Z G 0 G and Z G 1 GH. The other free parameters of Z G M will be derived through collocation. Using (.6) and (.7) equation (.1) can be rewritten to GL G M 9 NZ G M 9, G,L G M 9 X G M 9 (.8) The performance of the collocation method depends heavily on the discretization points in each interval used for the Lagrangian polynomial. These can be derived by using Gaussian quadrature formulas [6]. The first tests with the numerical optimization algorithms have shown that it is most efficient to have the boundaries 0 and 1 included in the discretization scheme, which e.g. is given by the Gaussian-Lobatto quadrature formulas. Finally, the objective function can be evaluated using the same quadrature formula: IBC G, G,;< G L G M, G,L G MM 9 <;<Z G M 9, G,L G M 9 9F where [ 9 and M 9 are the weights and supporting points of the Gaussian quadrature formula. The derived general discretized model description for the NOC with collocation is finally stated by:,, 7 78 ;<@@[ 9 <\ G L G M 9, G,L G M 9 ] subject to GF 9F Z G 0 G, Z G 1 1 Z G M^, G,L G M^X G M^ (.9) (.10) (.11) G _" #3 G, GH 5 (.1) G, G, 0 and (.13) 7, 78, 0 " 7 0 (.1) (.15) for 0,>,J1 and `1,>,aJ, where a is the degree of freedom from the basis function. 5 Modelica Applications To investigate the performance of the proposed optimization algorithm, several optimal control problems are solved and corresponding results are presented in this section. Additional examples will be presented in the final paper. 5.1 Batch Reactor We begin by considering a simple model from the chemical reactor described in [8] to maximize the yield of c by manipulation the reaction temperature, with the following problem formulation:,,j c1 (5.1) subject to d Jec f< g < 1J j n c i1j c m h5jl (5.) 0 (5.3) (5.) where, c o and #30,15.

5 11 3 J 1 1 J "Oƒ 1 0 "Oƒ 0.8u amb u amb u ˆ z 3u ˆ 300"Oƒ "Oƒ 0 1, 1 Figure. Trajectories of state and control variables 5. Optimal control of Diesel-Electric powertrain The Diesel-electric model based on [11] is presented in Appendix A. This concept is modeled according to a nonlinear mean value engine model (MVEM) containing four states and three control inputs while the generator model is simplified by considering constant efficiency and maximum power over the entire speed range. In a Diesel-electric powertrain the operating point of the Diesel engine can be freely chosen which would potentially decrease fuel consumption. Moreover, the electric machine has better torque characteristics. These are the main reasons making the Diesel-electric powertrain concept interesting for further studies. To investigate the fuel optimal transients of the powertrain from idling condition to a certain power level while the accelerator pedal position is interpreted as a power level request, the following optimal control problem is solved: and boundary conditions are: at 0,d at, c z { c z { g idle operating values, 0, d c z { gdesired values, and z u E š. The constraints are originated from components limitations and E functions are described in [11]. In this work, we try to find the fuel optimal control and state trajectories in a certain time interval [0,0.5]. For simplicity, only diesel operating condition is assumed which means ( z u tq7 0). The dynamic system is solved after it is discretized into subintervals. Figure 5 and Figure 6 show the obtained control and state trajectories. As it is expected, the fuel optimal results happen when engine is accelerated only near the end of the time interval (N0.3 ƒ) to meet the end constraints while minimizing the fuel consumption. In section 6 it is shown how the parallel execution increases the performance of the optimization process. [ Gpq O G O q States d g, Controls r st v [ u tq7 p y min subject to c c, z,, z c z, c, { z {, c, z,, c { c, z, {, c } c, { J ~, c 0 ~, c, J z J z, z 0 Figure 5 Trajectories of control variables

6 NSI 1 thread threads threads 500 3,158,61,06 6 threads, ,39 6,61,867, ,076 1, 1,15 1, ,671,06,871 0, ,656 86,86 50,39 1, ,065 11,07 80,6 61,5 Figure 6 Trajectories of state variables 6 arallel Execution and erfor- mance Measurements 6.1 arallel Execution using Multiple Shooting We have performed measurements for different dis- of integrator cretization intervals and different number steps in each interval. The C++ source code is compiled by gcc version.6.3 (GCC) with OpenM support. The measurements are done on an Intel Xeon E550 CU with 16 GHz each. The corresponding optimization problem is solved by the interior point optimizer Ipopt [19]. The speedups obtained and the computation times for the Diesel model are shown in Figure 7. NSI, No. Integrator steps 1 thread threads 8 threads threads 100,1 661,18 05,15 57, ,6 169,575 50, 37,16 0,555 11,61 7,,13 6,101 5, 0,558 11,359 6,515,,071 3,368 1,8 3,7,007 1,108 0,,881 0,689 Speedup Diesel Model Speedup 100,1 50, 5, 1,8 NSI, Integrator steps Figure 7. Speedups and computation times 16 threads Threads Threads 8 Threads 16 Threads The results for the Batch reactor model are shown in Figure 8. Figure 8. Speedups and computation times 6. arallel Execution using Collocation Currently, a sequential implementation of the NOC with collocation has been achieved and tested. arallel execution using this discretization scheme will be im- will be demon- plemented and corresponding results strated in the final paper. 7 Integration with OpenModelica Support for specifying optimization goal functions and constraints together with Modelica models has now been implemented in OpenModelica. Such integrated models can now be exported via XML to tools such as CasADi [1] which can act as a frontend to ACADO [15]. In the current OpenModelica prototype all aspects of the tool chain are not yet completely implemented. For example, we are currently using numerical Hessi- machinery in ans since the automatic differentiation OpenModelica has not yet been extended to operate on the optimization problem goal function. However, the prototype is complete enough to do the measurements of the included model applications on a parallel platform to obtain the speedup curves for parallel execution on 1-16 cores. In the final version of the paper we expect to have completed the OpenModelica tool-chain integration. The OpenModelica compiler has been extended to export Modelica Models to XML based on an extended version of the FMI XML schema from [16]. The XML export, in addition to the standardd Modelica syntax, supports the Optimica extensions from Jmodelica [18]. Theses extensions allow users to formulate dynamic

7 optimization problems to be solved by a numerical algorithm. The extensions include several constructs including a new specialized class optimization, a constraint section, etc. See the batch reactor example below as well as the Optimica manual for complete information. The XML generated for flattened Optimica Models can be imported into other non-modelica Optimization tools like ACADO. optimization BatchReactor (objective = -x(finaltime), starttime = 0, finaltime =1) Real x1(start=1,fixed=true,min=0,max=1); Real x(start=0,fixed=true,min=0,max=1); input Real u(free=true, min=0, max=5); equation der(x1) = -(u+u^/)*x1; der(x) = u*x1; end BatchReactor; Currently the OpenModelica compiler does not yet use the optimization problem formulation internally as input to automatic differentiation. The Modelica plus Optimica model description is flattened, some common compilation phases are applied e.g. syntax, semantics and type checking, simplification, constant evaluation etc. and then the complete flat model is exported to XML. For the final paper the compiler will be able to process and analyze optimization specific information internally. Especially, derivative information (Jacobian, Hessian) will be generated symbolically[13] and given to the optimization algorithm. 8 Conclusions Nonlinear model predictive control (NMC) has become increasingly important for today s control engineers during the last decade. In order to apply NMC a nonlinear optimal control problem (NOC) must be solved which needs a high computational effort. State-of-the-art solution algorithms are based on multiple shooting and/or collocation algorithms, which are needed to solve the underlying dynamic model description. In the final paper we will have presented and compared parallel implementations of multiple shooting and collocation, which lead to a very fast solution of the underlying NOC suitable for on-line applications. The modeling and problem description is done in Modelica with the Optimica language extension. The simulation is performed using OpenModelica. Speedup curves for parallel execution are presented. To our knowledge this is one of the first designs and implementations of parallel multiple shooting and collocation algorithms. 9 Acknowledgements This work has been partially supported by Serc, by SSF in the EDOp project and by Vinnova as well as the German Ministry BMBF (BMBF Förderkennzeichen: 01IS0909C) in the ITEA OENROD project. The Open Source Modelica Consortium supports the OpenModelica work. References [1] Open Source Modelica Consortium. OpenModelica System Documentation Version 1.8.1, April [] Modelica Association. The Modelica Language Specification Version 3., March th Modelica Association. Modelica Standard Library 3.1. Aug [3] Jasem Tamimi, u Li. A combined approach to nonlinear model predictive control of fast systems. Journal of rocess Control, 0, pp , 010. [] Biegler, Lorenz T Nonlinear rogramming: Concepts, Algorithms, and Applications to Chemical rocesses. s.l. : Society for Industrial Mathematics, 010. [5] Munz, Claus-Dieter and Westermann, Thomas Numerische Behandlung gewöhlicher und partieller Differenzialgleichungen. Berlin Heideberg : Springer Verlag, 009 [6] Stoer, Josef Einführung in die Numerische Mathematik I. Berlin Heideberg : Springer Verlag, [7] Heuser, Harro Gewöhnliche Differentialgleichungen. Wiesbaden : Teubner Verlag, 006. [8] Tamimi, Jasem Development of Efficient Algorithms for Model redictive Control of Fast Systems. Düsseldorf: VDI Verlag, 011. [9] Friesz, Terry L Dynamic Optimization and Differential Games. US: Springer US, 007. [10] Folkmar, Bornemann und Deuflhard, eter Numerische Mathematik: Numerische Mathematik : Gewöhnliche Differentialgleichungen: Bd II: [Band]. s.l. : Gruyter, 008. [11] Martin Sivertsson and Lars Eriksson Optimal power response of a diesel-electric powertrain. Submitted to ECOSM 1, aris, France, 01. [1] etzold L. R.: A Description of DASSL: A Differential/Algebraic System Solver, Sandia National Laboratories Livermore, 198.

8 [13] Braun, Willi, Ochel Lennart and Bachmann Bernhard. Symbolically Derived Jacobians Using Automatic Differentiation - Enhancement of the OpenModelica Compiler, Modelica Conference 011 [1] Joel Andersson; Johan Åkesson; Moritz Diehl, CasADi - A symbolic package for automatic differentiation and optimal control, roc. 6th International Conference on Automatic Differentiation, 01. [15] Houska, B., Ferreau, H.J., and Diehl, M. (011). ACADO toolkit - an open source framework for automatic control and dynamic optimization. Optimal Control Applications & Methods, 3(3), [16] Functional Mock-up Interface: [17] aul Heckbert. Introduction to Scientific Computing. notes/ode/bvp.html [18] Johan Åkesson. Optimica An Extension of Modelica Supporting Dynamic Optimization. In 6 th International Modelica Conference 008. Modelica. Association, March 008 [19] Interior oint OTimizer (Ipopt)

9 10 Appendix A Diesel Engine Model owertrain model [ Gpq u GpqJu tq7 [ Gpq tq7 q Intake System Compressor Π p Iž Ÿž, Π p, žÿ& cp y Ÿž 1 p, ±± Ÿž / ±³, u y Ÿž /y p p Ÿ y Ÿž, µ ±³ % p Q p Intake manifold Iž Cylinder Gas Flow Ÿy Iž I pjpg, G p V ªŸ ªŸ«C, p,p p,p, 1J ªŸ ªŸ«C 8 pg % ¹ º Iž I ³» {¼ Ÿ y Iž, µ ½ V Q ½ V, «¾ {¼ [ Gpq µ p V, À I Torque Á/ Qà J "Q J OO, OO Ä Å O Æ Ã JO, J6 10 QÈ; É ÊÄ µ, Æ 1 µ Gt µ Gt,pË 1J Ì " ͺ 8 p "Q Ä Å Æ 105 Q "1 d [ 60 QÃÆ 1000 g Q " d [ 60 QÃÆ 1000 g Q "3,žŸ& c,

10 Temperature Π p ³ž, q G7 Ï ÐÑ Iž H Ÿ, Ò 1 Ï IÓ ¹ p ¹Ÿ y Iž ªŸ«C q µ p Π q 8 Ì Ÿ "p 8Ì Ÿ Ì Ÿ 8 eqg7 e 1J p½ Q p½ Q ½ f G " p Ì Ÿ 8 f Exhaust System Exhaust Manifold ³ž Turbine ³y ³ž ³ž pgjj st, q q Π ³, Π Ô Õ e Π, c ³ž Ì ³ 8 ª³ ª³«C f, Z Π Ô Ö cì ³ \Π Ì ³ 8 Ô ª³ Ô JΠ ª³BC ª³ ], ª³«C ³ž Z ³ y Á,q, u Q q q µ \1JΠ ª³ ], µ Q, p 8 c J[ ³ž Gp [ p Wastegate Π st ³ Ô, Π st Õ eπ st, c ³ž st O q Ø q q Z st st Á st,q Ì ³ H ª³ Ô ª³«C f, Z st Ö cì ³ \Π ª³ Ì ³ 8 st JΠst ª³BC Ô ª³ ] Model Constants Symbol Description Value Unit Ù ÚÛÜ Ambient pressure e5 a Ý ÚÛÜ Ambient temperature 98.6 K Þ ÙÚ Specific heat capacity of air, constant pressure 1011 J/(kg.K) Þ ßÚ Specific heat capacity of air, constant volume 7 J/(kg.K) à Ú Specific heat capacity ratio of air á Ú Gas constant, air 87 J/(kg.K) Þ Ùâ Specific heat capacity of exhaust gas, constant pressure 133 J/(kg.K) à â Specific heat capacity ratio of exhaust gas á â Gas constant, exhaust gas 86 J/(kg.K) à Þãä Specific heat capacity ratio of cylinder gas Ý åû Intake manifold temperature K Ù âæ ressure in exhaust system e5 a ç/è æ Stoichiometric oxygen-fuel ratio é êë Diesel heating value.9 e6 J/kg

11 Model arameters Symbol Description Value Unit ì Þãä Number of cylinders 6 - ë í Engine displacement z î Þ Compression ratio ï ðâìæâñ Inertia of the engine-generator 3.5 ` c ë åæ Volume of intake system z á Þ Compressor radius 0.0 m ò ÛÚó Max. compressor head parameter Û Þ,Þôîî,ÛÚó Max. corrected compressor mass flow 0:56 - õ Þ Compressor efficiency J/(kg.K) õ ßôä Volumetric efficiency õ åð,þö Combustion chamber efficiency Þ îø Friction coefficient 8.1 e-5 - Þ îù Friction coefficient e-3 - Þ îú Friction coefficient õ æþ Non-ideal Seliger cycle compensation ó Þß Ratio of fuel burnt during constant volume ë âû Volume of exhaust manifold z ï ñþ Turbocharger inertia e- ` c û îåþ Turbocharger friction.358 e-5 ` c /"Õ ç ñ,â Effective turbine area e- z õ ñ Turbine efficiency Þ üð,ø Wastegate parameter Þ üð,ù Wastegate parameter ç üð,â Effective wastegate area e- z