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Model and discretization impact on oscillatory optimal control for a diesel-electric powertrain
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IFAC-PapersOnLine 48-15 (2015) 066–071 Model and discretization impact on Model and discretization impact on oscillatory optimal control for aon Model and discretization impact oscillatory optimal control for a diesel-electric powertrain oscillatory optimal control for a diesel-electric powertrain diesel-electric Martin Sivertsson ∗ powertrain Lars Eriksson ∗
PPgen P [kW] [kW] P[kW][kW] gen gen gen
2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 66 Hosting by Elsevier Ltd. All rights reserved. Peer review© of International Federation of Automatic Copyright 2015 IFAC 66 Copyright ©under 2015 responsibility IFAC 66 Control. 10.1016/j.ifacol.2015.10.010
gen gen gen gen
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Eriksson ∗ ∗ Eriksson Eriksson ∗ Link¨ Engineering, oping Engineering, Link¨ oping {marsi, larer}@isy.liu.se. Vehicular Systems, Dept. of Electrical Engineering, Link¨ oping ping University, SE-581 83 Link¨ oping, Sweden, Engineering, {marsi, larer}@isy.liu.se. Vehicular Systems, Dept. of Electrical Link¨ o University, o University, SE-581 SE-581 83 83 Link¨ Link¨ oping, ping, Sweden, Sweden, {marsi, {marsi, larer}@isy.liu.se. larer}@isy.liu.se. Abstract: A mean value engine model is used to study optimal control of a diesel-electric Abstract: mean value engine is used studytooptimal control of a diesel-electric powertrain. AThe resulting optimalmodel controls are to shown be highly oscillating for certain Abstract: A mean value engine model is to study optimal control of powertrain. resulting optimal controls arethis shown be highly oscillating formodeling certain Abstract: AThe mean valuethe engine model is used used to study optimal control of aa diesel-electric diesel-electric operating points, raising question whether is antoartifact of discretization, powertrain. resulting optimal controls are shown be oscillating for certain operating raising the question whether is anto of discretization, powertrain. The resulting optimal controls arethis shown toartifact be highly highly oscillating formodeling certain choices or points, a The phenomenon available in real engines. Several model extensions are investigated operating points, raising the question whether this is an artifact of discretization, modeling choices or a phenomenon available in real engines. Several model extensions are investigated operating points, raising the question whether this is an artifact of discretization, modeling and their corresponding optimal control trajectories are studied. It is shown that the oscillating choices or a phenomenon phenomenon available in real real engines. Several model extensions arethe investigated and their corresponding optimal control trajectories are studied. It is shown that oscillating choices or a available in engines. Several model extensions are investigated controls cannot be explained by the implemented extensions to the previously published model, and corresponding optimal control are studied. It shown that the oscillating controls cannot be explained by the implemented extensions to the previously published model, and their corresponding optimal control trajectories are studied. It is is shown that the solution oscillating nor their by the discretization, showing thattrajectories for certain operating points the optimal is controls cannot be by extensions to previously published model, nor by the discretization, that for certain operating points the optimal solution is controls cannot be explained explainedshowing by the the implemented implemented extensions to the the previously published model, periodic. nor by by the the discretization, discretization, showing showing that that for for certain certain operating operating points points the the optimal optimal solution solution is is periodic. nor © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. periodic. periodic. 200 1. INTRODUCTION 2400 200 9 1 2 3 4 5 6 10 8 11 1. INTRODUCTION 7 2400 200 9 4 5 6 10 8 11 1. INTRODUCTION 2000 1 2 3 7 200 1. INTRODUCTION 2400 The engine speed of a conventional vehicle is normally 2400 9 4 10 8 11 2000 1 7 9 1 2 2 3 4 5 5 6 6 10 3 8 11 7 The engine speed of a conventional vehicle is normally 100 decided by the wheel speed and the gear ratio. In a diesel1600 2000 2000 The engine speed of speed conventional vehicle isInnormally normally 100 decided by the wheel and thepath gear ratio.is a dieselThe engine speed of aamechanical conventional vehicle 1600 electric powertrain this between the com1200 100 decided by the wheel wheel speed and is thereplaced gear between ratio. In diesel1600 100 electric mechanical path comdecided by the speed and the gear ratio. aaelectric diesel1600 bustion powertrain engine andthis the wheels by In anthe 1200 electric powertrain this mechanical path between the combustion engine and the wheels is replaced by an electric 800 electric powertrain this mechanical path between the com0 path instead. This introduces an extra degree of freedom 1200 200 400 600 800 1000 1200 1400 1600 1800 1200 bustion engine and the can wheels iscontrolled replaced by of an freedom electric 800 0 time [s] path instead. This introduces extra degree bustion and the wheels replaced by an electric since theengine engine speed beanis independently 200 400 600 800 1000 1200 1400 1600 1800 800 0 time [s] path instead. This introduces ancontrolled extra degree of freedom freedom 0 since the engine speed can offers bean independently path This introduces extra degree of 200 400 600 800 1000 1200 1400 1600 1800 WHTC and its traction phases. of theinstead. wheel speed, which the potential of both Fig.8001. The 200 400 600 800 1000 1200 1400 1600 1800 time [s] [s] since the engine speed can offers be controlled independently 150 traction phases. of thethe wheel which the potential of both since engine speed can be independently optimizing thespeed, performance andcontrolled consumption since the Fig. 1. The WHTC and its time 20001. The is 150 hypothesis that the oscillations seen in the optimal Fig. WHTC and its traction phases. of the wheel speed, which offers the potential of both optimizing the performance and consumption since the Fig. 1. The WHTC and its traction phases. of the wheel speed, which offers the potential of both operating point of the diesel engine can be controlled more 2000 150 hypothesis is that the oscillations seen in the optimal 150 100 optimizing the performance and consumption since the variable geometry turbine (VGT) control of a diesel engine operating the diesel engine can be controlled more optimizing the performance and consumption the freely thanpoint in aof conventional powertrain. This since of course 2000 hypothesis is that the oscillations seen in the optimal 2000 100 is variable geometry turbine (VGT) control of a diesel engine hypothesis is that the oscillations seen in the optimal 1500 are due to decrease in the gas exchange losses. This operating point of the diesel engine can be controlled more freely in aofconventional This degree of course operating theofdiesel be extra controlled more raises than the point question how engine topowertrain. usecan this of variable 100 geometry turbine (VGT) control of aa diesel engine 1500 are due to decrease in the gas exchange losses. This is 100 variable geometry turbine (VGT) control of diesel engine 50 due to that the exhaust manifold pressure oscillates with freely than in a conventional powertrain. This of course raises the question of how to use this extra degree of freely than in a conventional powertrain. This of course freedom. 1500 are due to decrease in the gas exchange losses. This is 50 1500 due to that the exhaust manifold pressure oscillates with are due to decrease in the gas exchange losses. This is raises the the question question of of how how to to use use this this extra extra degree degree of of the1000VGT position whereas the intake manifold pressure freedom. raises 50 due to that the exhaust manifold pressure oscillates with 50 1000 the VGT position whereas the intake manifold pressure 0 Previously it has been studied how to optimally control the due to that the exhaust manifold pressure oscillates with remains unaffected due675to the turbocharger dynamfreedom. 655 660 665 670 680 slower 685 690 695 700 705 freedom. time [s] intake manifold pressure VGT position whereas the Previously has been studied howpower to optimally control the the 1000 remains unaffected due to the slower turbocharger dynampowertrain itbetween two different levels, see Sivertsthe VGT whereas the intake manifold pressure ics.1000 This could that optimal solution is in0 fact 655 660position 665indicate 670 675 680 685 690 695 700 705 time [s] Previously it has been studied how to optimally control the remains unaffected due to the slower turbocharger dynampowertrain between two different power levels, see SivertsPreviously it has been studied how to optimally control the ics. This could indicate that the optimal solution is705 in00 fact son and Eriksson [2015a,b]. For off-highway machinery remains unaffected due675 to slower turbocharger dynam655 660as 665 665 670 675 680 685 690 695 Gilbert 700 705 periodical described inthe Gilbert [1977]. 655 660 670 680 690 695 700 time [s] 685 [1976], time [s] powertrain between two different power levels, see Sivertsics. This could indicate that the optimal solution is in fact fact son and patterns Eriksson [2015a,b]. For off-highway machinery the periodical powertrain between different power levels, see Sivertsas described in Gilbert [1976], Gilbert [1977]. driving aretwo normally very transient, something ics. This could indicate that the optimal solution is in Other possible explanations are either that the solution is son and Erikssonin[2015a,b]. [2015a,b]. For off-highway machinery the Other periodical as described described in that Gilbert [1976], Gilbert [1977]. driving patterns are very transient, something son Eriksson For off-highway machinery the explanations are the either that Gilbert the is thatand is captured thenormally World Harmonized Transient Cycle periodical as Gilbert [1976], [1977]. along apossible singular arc andin controls aresolution therefore driving patterns are normally very transient, something explanations are either that the solution is that is captured inare thenormally World Harmonized Transient Cycle driving patterns very transient, something along apossible singular arc and controls are therefore (WHTC), see WHDC Working Group [2005], shown in Other Other possible explanations are the either that or the solution is oscillatory, as discussed in that Schwartz [1996], that it is an that is captured in the World Harmonized Transient Cycle along a singular arc and that the controls are therefore (WHTC), see WHDC Working Group [2005], shown in that is captured in the World Harmonized Transient Cycle as discussed in that Schwartz [1996], orare that it is an Fig. 1. The WHTC can be divided into 11 traction phases, oscillatory, along a singular arc and controls integration error exploited by the algorithm to therefore decrease (WHTC), see WHDC Working Group [2005], shown in integration oscillatory, as discussed inHellstr¨ Schwartz [1996], or that that it both is an an Fig. 1. The WHTC can be divided into 11[2005], traction phases, (WHTC), see WHDC Working shown in error exploited by the to decrease defined as the period between twoGroup idle periods, where idle oscillatory, Schwartz [1996], or it is the criteriaas asdiscussed shown inin om algorithm et al. [2010]. In Fig. 1. The WHTC can be divided into 11 traction phases, integration error exploited by the the algorithm to decrease decrease defined as the period between two into idle periods, idle the Fig. 1. The WHTC can bethe divided 11 traction criteria as would shown in suspected Hellstr¨ om algorithm et al.the [2010]. In both is assumed to occur when engine speed is the where idlephases, speed integration exploited by to these cases error it be that frequency of defined as the the period between two idle periods, where idle the criteria as shown in Hellstr¨ oom et [2010]. In is assumed to occur when the engine speed is the where idleengine speed defined as between two idle periods, idle these cases it be suspected frequency of and no power isperiod required. Here the phases where the the oscillations criteria as would shown inon Hellstr¨ m that et al. al.the [2010]. In both both depend the discretization. To test this is assumed assumed toi.e. occur when the engine speed is the the the idle speed these cases would be suspected that the of and no power is required. the phases is to occur the speed is idle speed oscillations depend thepoint discretization. To this motored, Pgenwhen < 0 Here are engine ignored andwhere P inengine those the these cases ait itsingle would beon suspected that the frequency frequency of hypothesis operating is selected andtest studied gen is and noset power is required. required. Here the phases phases where theinengine engine the oscillations depend on the discretization. To test this are ignored andwhere Pgen is those hypothesis is motored, Pgen < 0 Here and no power is the the a single operating point is selected and studied cases to i.e. zero. the oscillations depend on the discretization. To test this using a very fine time discretization. is motored, motored, i.e. Pgen hypothesis single operating point is is selected selected and and studied studied gen < gen is cases set to i.e. zero.P <0 0 are are ignored ignored and and P Pgen is in in those those using is a veryaa fine time discretization. hypothesis single operating point To investigate the potential of the diesel-electric power- using a very fine time discretization. cases set to to zero. zero. cases set using a very fine time discretization. To investigate potential of the extra diesel-electric power2. CONTRIBUTIONS train and how the to best exploit degree of freeTo potential of diesel-electric power2. CONTRIBUTIONS train and how the tobybest degree of freeTo investigate the potential of the the extra diesel-electric powerdominvestigate introduced the exploit electrification of the powertrain, 2.ofCONTRIBUTIONS CONTRIBUTIONS train and how to best exploit the extra degree of freedom introduced by the electrification of the powertrain, train and how to best exploit the extra degree of freeminimizing fuel for the WHTC is cast as an optimal The contributions2. this paper is a deeper study of the dom introduced introduced by the electrification of the the powertrain, minimizing fuel by for theelectrification WHTC is cast as powertrain, an optimal The contributions of this paper for is adiesel-electric deeper study powerof the dom the of control problem (OCP). In a conventional powertrain occurence of oscillating controls minimizing fuel the WHTC is an optimal of this paper is aadiesel-electric deeper study of the control problem (OCP). In a speed conventional powertrain occurence oscillating controls minimizing fuel for for the engine WHTC is cast cast as an optimal The contributions of for this paper for iscontrol deeper study powerofMore the WHTC prescribes both and as output power, The trainscontributions as aofsolution optimal problems. control problem (OCP). In a conventional powertrain occurence of oscillating controls for diesel-electric powerWHTC both isengine output power, as aofit solution optimalthe control problems. More control problem (OCP). In a speed conventional powertrain occurence oscillating controls for diesel-electric powerbut hereprescribes engine speed a degree ofand freedom and also trains specifically studiesforwhether observed oscillations WHTC prescribes bothisisengine engine speed and output power, trains as solution for optimalthe control problems. More but hereprescribes engine a degree ofand freedom also specifically studies whether observed oscillations WHTC both speed power, trains aa it solution optimal control problems. More optimized. If this speed OCP solved for phase 8 output in theand WHTC are an as artifact of thefor discretization. It also investigates if but resulting here engine engine speed issolved a very degree of freedom freedom and also specifically it studies studies whether the observed oscillations optimized. If this OCP isare foroscillatory, phase 8 in see theand WHTC an artifact of thebediscretization. It investigates if but here speed is a degree of specifically it whether oscillations the controls Fig.also 2, are the oscillations can explainedthe by observed thealso models used and optimized. If OCP for phase in the an artifact of the It investigates if the resulting controls Fig.and 2, are the oscillations can explained byimpacts thealso models used and optimized. If this this OCP isare solved foroscillatory, phasein8 8Sivertsson in see the WHTC WHTC are an artifact of thebediscretization. discretization. It also investigates if t ∈ [670, 678], [684, 687].is Itsolved isvery mentioned whether or not extending the model the oscillating the resulting controls are very oscillatory, see Fig. 2, the oscillations can be explained by the models used and tEriksson ∈ [670, 678], [684, 687]. It is mentioned in Sivertsson and whether or not extending the model impacts the oscillating the resulting controls are optimal very oscillatory, seetransient Fig. 2, the oscillations can be also explained by the models and [2015a] that the solutions in solutions. The paper presents a fast and used accurate ∈ [670, [670, control 678], [684, 687]. It is is mentioned in Sivertsson Sivertsson and solutions. whether or not extending thepresents model impacts the oscillating Eriksson [2015a] the optimal solutions in are transient The paper also fastoptimal andoscillating accurate tt ∈ 678], [684, 687]. It mentioned in and whether not extending the model the optimal ofthat a diesel-electric powertrain often residual or gas model suitable for use impacts ina an control Eriksson control [2015a] that the optimal optimal solutions in are transient solutions.gas The paper also presents presents fastoptimal and accurate accurate optimal a diesel-electric powertrain often residual model suitable for use inaa an control Eriksson [2015a] the solutions in transient solutions. The paper also fast and oscillatory and inofthat Asprion et al. [2014] the unconfirmed context. optimal aa diesel-electric powertrain are oscillatory and inof et al. [2014] the unconfirmed context. optimal control control of Asprion diesel-electric powertrain are often often residual residual gas gas model model suitable suitable for for use use in in an an optimal optimal control control oscillatory and in Asprion et al. [2014] the unconfirmed context. oscillatory and in Asprion et al. [2014] the unconfirmed context. Copyright © 2015 IFAC 66 ωω [rpm] ω[rpm] ω[rpm] [rpm]
Martin Sivertsson ∗ Lars ∗ ∗ Lars Martin ∗ Martin Sivertsson Sivertsson Lars Vehicular Systems, Dept. of Electrical ∗ Vehicular Systems, Dept. of Electrical University, SE-581 83 Link¨ o ping, Sweden, ∗ ∗
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Fig. 2. The optimal solution to phase 8 of the WHTC, with ωice as a degree of freedom. The resulting controls are highly oscillatory, see t ∈ [670, 678], [684, 687] Description Pressure Temperature Rotational speed Massflow Power Torque Energy Pressure ratio Volume Specific heat capacity ratio Specific heat capacity constant pressure Specific heat capacity constant volume Gas Constant Control signals Inertia Blade speed ratio Fuel-air equivalence ratio Air-fuel smoke-limit Residual gas fraction Fuel to mass ratio Lower heating value Compression ratio
Index GenSet em ac wg tc e mech c, surge d pump eo
Description Engine-Generator Exhaust manifold After compressor Wastegate Turbocharger Exhaust Generator-mechanical Compressor surge-limit Displaced Pumping Engine out
4. PROBLEM FORMULATION
Table 1. Symbols used Symbol p T ω m ˙ P M E Π V γ cp cv R uf , uwg , Pgen J BSR φ λmin xr MF R qHV rc
Description Engine Intake manifold Compressor Turbine Fuel Air Generator-electrical Reference Volumetric Friction Indicated gross Ambient
turbocharger speeds, ωice/tc , and inlet and exhaust manifold pressures, pim/im . The controls are injected fuel mass, uf , wastegate position, uwg , and generator power, Pgen . The engine model consists of two control volumes, intake and exhaust manifold, and four restrictions, compressor, engine, turbine, and wastegate. The governing differential equations of the MVEM are: dωice Pice − Pmech = (1) dt ωice JGenSet dpim Ra Tim = (m ˙ c−m ˙ ac ) (2) dt Vim dpem Re Tem = (m ˙ ac + m ˙ f −m ˙ t−m ˙ wg ) (3) dt Vem dωtc Pt ηtm − Pc = (4) dt ωtc Jtc For a complete list of the symbols used in the paper, see Table 1-2.
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Index ice im c t f a gen ref vol f ric ig amb
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Table 2. Subscripts used
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This paper uses the MVEM to study optimal stationary operation, or lack of it in the case of oscillating controls.
Unit Pa K rad/s kg/s W Nm J m3 J/(kg · K) J/(kg · K) J/(kg · K) mg/cycle, -, W kg · m2 J/kg -
4.1 Stationary optimization As a reference for the dynamic optimization, three stationary optimization problems are first solved, to find the following three stationary points for the given ωref , Pref combination: The maximum efficiency, φmax , the maximum fuel/air-ratio, ηmax , and the minimum fuel/air-ratio, P φmin . η = m˙ fgen qHV is the efficiency of the powertrain and φ is the fuel/air-ratio. These problems are solved to find the optimal operating point for stationary operation and also the limits for stationary operation. 4.2 Dynamic optimization The main optimal control problem studied is: T m ˙f min
3. MODEL
u(t)
0 (5) x(t) ˙ = f (x(t), u(t)) (x(t), u(t)) ∈ Ω(t) where x is the state vector of the MVEM, x˙ is the state equations (1)-(4), and u = [uf , uwg , Pgen ]. The optimal control problems are also subject to a set of constraints, namely:
The basic model used can be downloaded in the LiU-DEl-package from systems software [2014] and is described in detail as M V EMo in Sivertsson and Eriksson [2014]. The modeled diesel-electric powertrain consists of a 6cylinder diesel engine with a fixed-geometry turbine and a wastegate for boost control, with a generator mounted on the output shaft. The states of the MVEM are engine and
s.t.
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ωice [rad/s]
x(0) = x(T ) = x(ηmax ), x(T ˙ )=0 xmin ≤ x(t) ≤ xmax umin ≤ u(t) ≤ umax , ωice (T ) = ωref or ωice (t) = ωref , Pgen (t) = Pref 1 Pice (x(t), u(t)) ≤ Pice,max (x(t)), φ(x(t), u(t)) ≤ λmin BSRmin ≤ BSR(x(t), u(t)) ≤ BSRmax , Πc ≤ Πc,surge (6) The constraints are actuator and state limits, as well as constraints imposed by the components, such as maximum power of the engine, Pice , surge-limit of the compressor, Πc,surge , blade speed ratio-limit of the turbine, BSR, as well as environmental constraints, i.e. an upper limit on φ set by the smoke-limiter.
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The driving mission-constraints are that the powertrain starts in the operating point of maximum efficiency ηmax , a point it should also end in, with the added requirement that the end operating point should be stationary. The generator power is also fixed to the reference value. Two types of problems are then studied, one where the engine speed is fixed to the reference speed, denoted ωice = fix, and one where it is allowed to depart from this as long as it starts and ends in ωice = ωref , denoted ωice = free.
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5. NUMERICAL SOLUTION The software package that is used to solve the optimal control problem numerically is CasADi Andersson [2013]. First the problem is discretized using Radau collocation with three collocation points in each control interval. The states are thus approximated with a third order polynomial, whereas the controls are approximated by a second order polynomial in each control interval. The states are required to be continuous over each control interval boundary, whereas the controls are allowed to be discontinuous. The resulting nonlinear program(NLP) is solved using IPOPT, W¨ achter and Biegler [2006], with the MA57 linear solver from the HSL package, HSL [2013]. For the wastegate oscillation study 200 control intervals have been used.
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Fig. 3. Optimal states and controls for constant output power, both stationary and dynamic. The dynamic solutions are highly oscillatory. losses only depend on engine speed except the pumping torque, Mpump , see Appendix A.1. This means that the oscillating control actually decreases the pumping torque, as hypothesised in Asprion et al. [2014]. Looking at Fig. 3 the low pass filtering effect of the turbocharger can be clearly seen since the wastegate opening and closing results in a pem span of 60-100 kPa depending on if ωice = free/fix, whereas the effect on pim is only 4-5 kPa. In Fig. 4 the pumping power, Ppump = ωice Mpump , are shown relative Ppump (ηmax ). Both x(φmin ) and x(φmax ) increase Ppump compared to x(ηmax ), as expected. The oscillation changes Ppump several hundred percent away from what is stationary optimal.
6. OSCILLATING CONTROLS The stationary point of interest here is the one seen with oscillating controls in Fig. 2 namely Pgen = 140 kW and ωice = 1700 rpm. The three stationary OCPs (OSS) described in Section 4.1 and the two dynamic OCPs (OSD), ωice = fix and ωice = free, described in Section 4.2 are solved using CasADi/IPOPT/HSL and the results are shown in Fig. 3. Looking at Fig. 3 it is clearly visible that both OSDs result in a periodic oscillation. If the engine speed is free the amplitude and frequency changes slightly, however the nature of the oscillation remains the same. The nature of the opening and closing goes against the hypothesis that this should be oscillations due to a singular arc or an effect of the integration error since the wastegate only opens 9-11 times despite 200 control intervals with a three controls in each interval, yielding an optimal period of 90-110ms which is approximately 20 times the control interval length. This indicates that the oscillations are in fact optimal.
In Table 3 the gains of oscillating controls are quantified. x(φmax ) and x(φmin ) both increase the pumping energy Epump with 50% which also leads to a relative efficiency decrease of 0.5% (0.2% absolute). The oscillating control with fixed ωice decreases Epump with 2.4% and ωice free with 4.1%. Since the friction losses are quadratic in engine speed, ωice -free, increases the friction losses, but it is still beneficiary since the relative efficiency increase is 0.52 vs. 0.24 for ωice -fix (0.2 vs. 0.09 absolute). The gains are small but nevertheless surprising since it’s a dynamic phenomenon. Looking at Fig. 5 the wastegate’s effect on the stationary efficiency as well as pumping torque is shown. The efficiency is a convex function in uwg whereas Mpump is concave, which if the analysis was per-
This is especially interesting for the case with fixed engine speed, since then the effect can be isolated since all torque 68
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satisfies both conservation of mass and energy is implemented.
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The basic model is M V EMo which is then extended with different combinations of these three models. The model fit for the different models versus both dynamic(Dyn.) and stationary measurements(Stat) are shown in Table 4. None of the model extensions have any significant effects on the model fit versus measurements.
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Fig. 4. Pumping power relative the stationary optimal point. x(φmax ) x(φmin ) ωice = fix ωice = free
∆η -0.542 -0.482 0.024 0.052
∆Ef ric 0.0 0.0 0.0 0.8797
∆Epump 56.3782 50.1279 -2.4073 -4.0869
7.1 Residual gas In Wahlstr¨om and Eriksson [2011] the engine out temperature model is based on an ideal Seiliger cycle model, incorporating residual gas. The model is formulated as: xr,0 = 0,Teo,0 = 800. While Teo,k+1 − Teo,k >1e-6 m ˙ f qHV qin = (1 − xr,k ) m ˙ f +m ˙ ac qin xcv xp =1 + cv,a T1 rcγa −1 qin (1 − xcv ) xv =1 + qin xcv cp,a ( cv,a + T1 rcγa −1 ) (7) 1/γ −1/γ Πe a xp a xr,k+1 = rc x v T1 =xr,k+1 Teo,k + (1 − xr,k+1 )Tim
∆Eig 0.5448 0.4844 -0.0223 -0.0263
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Table 3. Changes in efficiency and energies T relative x(ηmax ) in percent, Ex = 0 Px dt. 0.375
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−10
[−]
a 1−γa 1/γa −1 Teo,k+1 =ηsc Π1−1/γ rc xp e xcv 1 − xcv γa −1 + qin + T 1 rc cp,a cv,a The equations in (7) are nonlinear and depend on each other and need to be solved using fixed point iterations. In Wahlstr¨om and Eriksson [2011] it is shown that if the solution from the previous time step is known, one iteration suffices to get a good approximation of the engine out temperature. In an optimization context it is difficult to keep track of the solution form the previous time step since the time steps are solved simultaneously. Also a submodel that is evaluated for a varying number of iterations is undesirable, especially since it complicates the computation of derivatives. Therefore this type of model is not implementable in an optimization context and a new model is developed.
Fig. 5. Wastegate position’s effect on the efficiency and pumping torque during stationary conditions. formed for stationary conditions would lead to the conclusion that oscillating controls would increase the pumping torque and consequently decrease the efficiency. However the result is actually the opposite, oscillating between the two worst controls from a stationary perspective, increases the efficiency dynamically. 7. MODEL EXTENSIONS To investigate whether the oscillating controls are results of a modeling assumption a set of different model extensions are considered. The extensions and their motivation are: • xr : Model for residual gas: In Sivertsson and Eriksson [2014] it is shown that mean/max absolute relative error increase of assuming xr = 0 is [0.014/0.06] versus measurements. However in the oscillating controls the exhaust pressure changes very rapidly, something that might have a significant effect on the amount of residual gas trapped in the cylinder. • ηvol,2 : Pressure ratio dependent volumetric efficiency model. In the volumetric efficiency model used, see Appendix A.2, only the dependence of the intake manifold pressure and engine speed are modeled. Of course the changing pressure ratio over the engine could have effects on the volumetric efficiency. • Adiabatic: The isothermal exhaust manifold model used, see Appendix A.3, neglects energy conservation since it assumes that the gases flowing in and out of the control volume have the same temperature, therefore an adiabatic exhaust manifold model that
xr is modeled as a function of both pressure ratio over the engine and the fuel to mass ratio. defined as: m ˙f (8) MFR = m ˙ f +m ˙ ac xr,mod1 cx Π2 + cxr2 Πe + cxr3 (9) = r1 e xr = xr,mod2 1 + cxr4 M F R In Fig. 6-top this gives a good agreement to the iterative model, resulting in mean/max relative errors of [0.92/4.52] %. Using that xcv becomes zero in the optimization for this particular engine Teo can be computed according to Teo =
MF R + Tim rcγa −1 ) (1 − xr )( qHVcp,a 1
1−1/γa 1−γa rc
ηsc Πe
− xr rcγa −1
(10)
Teo and xr could be used as starting values and then one iteration of the fixed point iteration can be performed. However it turns out that not only is it more computations, 69
IFAC E-COSM 2015 70 August 23-26, 2015. Columbus, OH, USAMartin Sivertsson et al. / IFAC-PapersOnLine 48-15 (2015) 066–071
0.14
xr /xr,m od1 [−]
1.4
0.1 0.08 0.06 0.04 0.02
1.2 1 0.8
Dyn. M V EMo xr xr +adiabatic ηvol,2 ηvol,2 + xr +adiabatic Stat. M V EMo xr xr +adiabatic ηvol,2 ηvol,2 + xr +adiabatic
0.6 0.4
0
0.2 0
1
2
3
4
0
0.02
Πe [−]
0.04
0.06
0.08
0.1
M F R [−] T :x
1200
eo
r, iter
T :x eo
r, mod
ωice T V 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 m ˙c Pc 2.5 1.8 2.5 1.8 2.5 1.8 2.5 2.0 2.5 1.9
pim T V 2.8 2.2 2.8 2.2 2.8 2.2 2.8 2.3 2.9 2.3 m ˙ ac Tem 2.5 2.4 2.4 2.5 2.5 2.7 2.7 2.3 2.7 2.6
pem T 2.8 2.8 2.8 2.9 2.9 m ˙ exh 3.3 3.3 3.1 3.2 3.0
ωtc V 2.9 2.9 3.0 3.0 3.0 Pt 5.4 5.5 4.9 5.5 4.9
T 2.9 2.9 2.9 3.0 3.1
V 2.9 2.9 3.0 3.2 3.2
+ Pmech 3.4 3.3 3.1 5.1 4.4
− Pmech 1.4 1.5 1.5 1.4 1.6
800
Ppump, rel [%]
MVEM 600 400 660
665
670
675
680
685
690
695
700
705
Ppump, rel [%]
655
rel error [%]
Ppump, rel [%]
Fig.0.056. Top: Residual gas, xr , vs. pressure ratio, Πe , and how this dependence is modeled (Left). The xr error’s dependence on M F R (Right). Bottom: Fit of the new 0 residual gas model for phase 8 of the WHTC. the−0.05 model fit is actually worse. When simulated over the entire WHTC the mean/max absolute relative error of −0.1 exhaust gas temperature, Teo for the model in (8)-(10) 655 660 665 675 680 685 690 700 705 relative the model in670(7) are [0.025/0.22]%. If 695 in addition to time [s] (8)-(10) one fixed point iteration is used the errors increase to [0.045/0.34] %. The model used is therefore the one without iterations. In Fig. 6-bottom the fit vs. the model in (7) is shown.
o
400 200 0 −200 −400
x
r
400 200 0 −200 −400
ηvol, 2
400 200 0 −200 −400
Ppump, rel [%]
Teo [K]
1000
400 200 0 −200 −400 −600
Ppump, rel [%]
xr [−]
Table 4. Mean absolute relative errors for the different models versus measurements. T=tuning set, V=validation set.
1.6
xr xr,m od1
0.12
400 200 0 −200 −400 −600
xr+adiabatic
η
+x +adiabatic
vol, 2
r
ωice=fix ωice=free 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 7. Relative pumping power of the optimal solutions to the two OSDs for the different models, relative the stationary optimal operating point. The oscillatory solution can not be explained by any of the model extensions.
7.2 ηvol,2 : Modified volumetric efficiency The ηvol,2 model implemented is a modified version of the model found in Eriksson and Nielsen [2014], consisting of an ideal part and two polynomials, in ωice and pim respectively. 1/γa rc − ppem im ηvol,ideal = rc − 1 2 ηvol,ωice =cηvol,1 ωice + cηvol,2 ωice + cηvol,3 ηvol =ηvol,ωice ηvol,ideal + cηvol,4 pim + cηvol,5 The component model fit for stationary measurements are slightly better than the model in (A.10), with mean/max absolute relative errors of 0.62/2.68 % vs. 0.9/3.7 % for (A.10), which is to be expected since the number of tuning parameters increases from three to five.
the manifold is computed according to (A.13), that is temperature after heat loss. 8. RESULTS The two OSDs defined in Section 4.2 are solved for the different models and the resulting optimal pumping power, Ppump is plotted relative the stationary optimal in Fig. 7. The oscillations cannot be explained by any of the model extensions. The periodic nature of the solution is present for all model extensions and the changes in frequency and amplitude of the oscillations are minor. In Table 5 the changes in energies and efficiency are shown, which confirms the results seen in Fig. 7. The decrease in pumping energy, ∆Epump , increases for each model extension for ωice = fix, indicating that given a standard mean value engine model it actually is optimal to use periodic wastegate control in order to decrease the pumping torque.
7.3 Adiabatic exhaust manifold model
The adiabatic model, as described in Chevalier et al. [2000] is implemented according to: dpem Re γe = (Tem,k (m ˙ ac + m ˙ f ) − Tem (m ˙ t+m ˙ wg )) dt Vem 9. CONCLUSIONS Re Tem dTem = ˙ ac + m ˙ f ) − Tem (m ˙ t+m ˙ wg ) γe Tem,k (m dt pem Vem Optimal control of a diesel-electric powertrain is studied. It is shown that the oscillatory solutions seen for certain − Tem (m ˙ ac + m ˙ f −m ˙ t−m ˙ wg ) operating points are not directly discretization dependent. This means extending the model with an additonal state, Instead the solution is periodic with a period much greater Tem . Tem,k , i.e. the temperature of the gases flowing into than the control interval length. Further it is seen that 70
IFAC E-COSM 2015 August 23-26, 2015. Columbus, OH, USAMartin Sivertsson et al. / IFAC-PapersOnLine 48-15 (2015) 066–071
Martin Sivertsson and Lars Eriksson. Optimal transient control trajectories in diesel-electric systems-part 1: Modeling, problem formulation and engine properties. Journal of Engineering for Gas Turbines and Power, 137(2), February 2015a. Martin Sivertsson and Lars Eriksson. Optimal transient control trajectories in diesel-electric systems-part 2: Generator and energy storage effects. Journal of Engineering for Gas Turbines and Power, 137(2), February 2015b. Vehicular systems software. ”http://www.fs.isy.liu. se/Software/”, 2014. Andreas W¨achter and Lorenz T. Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 106:25–57, 2006. Johan Wahlstr¨om and Lars Eriksson. Modelling diesel engines with a variable-geometry turbocharger and exhaust gas recirculation by optimization of model parameters for capturing non-linear system dynamics. Proceedings of the Institution of Mechanical Engineers, Part D, Journal of Automobile Engineering, 225(7):960–986, 2011. WHDC Working Group. Worldwide harmonized heavy duty emissions certification procedure. ”http:// www.unece.org/fileadmin/DAM/trans/doc/2005/ wp29grpe/TRANS-WP29-GRPE-50-inf04r1e.pdf”, 2005. GRPE-50-4-Rev.1, Read 2/2-2015.
Table 5. Changes in efficiency and energies of the two OSDs, ωice = fix and ωice = free, relative x(ηmax ), for the different models. M V EMo xr xr + adiabatic ηvol,2 ηvol,2 + xr + adiabatic
fix free fix free fix free fix free fix free
∆η 0.024 0.052 0.024 0.054 0.024 0.068 0.029 0.070 0.030 0.080
∆Ef ric 0.0 0.88 0.0 1.32 0.0 1.74 0.0 -2.23 0.0 -1.99
∆Epump -2.41 -4.09 -2.52 -3.64 -2.58 -3.7 -3.14 -10.16 -3.23 -10.21
71
∆Eig -0.02 -0.03 -0.02 0.02 -0.02 0.19 -0.03 -0.31 -0.03 -0.22
the pumping work of the engine decreases as a result of the oscillations. To study if this effect is a result of oversimplification in the previously published mean value engine model, several model extensions are investigated and their corresponding optimal control problems are solved. Furthermore a new residual gas model, suitable for optimal control, is presented. It is shown that the oscillating controls cannot be explained by the implemented extensions to the previously published model, showing that for certain operating points for mean value engine models the optimal solution is actually periodic. REFERENCES
Appendix A. EXCERPTS FROM ORIGINAL MODEL
Joel Andersson. A General-Purpose Software Framework for Dynamic Optimization. PhD thesis, Arenberg Doctoral School, KU Leuven, Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center, Kasteelpark Arenberg 10, 3001-Heverlee, Belgium, October 2013. Jonas Asprion, Oscar Chinellato, and Lino Guzzella. Optimal control of diesel engines: Numerical methods, applications, and experimental validation. Mathematical Problems in Engineering, 2014. Alain Chevalier, Martin M¨ uller, and Elbert Hendricks. On the validity of mean value engine models during transient operation. In SAE Technical Paper 2000-011261. SAE International, 2000. Lars Eriksson and Lars Nielsen. Modeling and Control of Engines and Drivelines. John Wiley & Sons, 2014. Elmer G. Gilbert. Vehicle cruise: Improved fuel economy by periodic control. Automatica, 12:159–166, 1976. Elmer G. Gilbert. Optimal periodic control: A general theory of necessary conditions. SIAM Journal of Control and Optimization, 15(5):717–746, 1977. Erik Hellstr¨ om, Jan ˚ Aslund, and Lars Nielsen. Design of an efficient algorithm for fuel-optimal look-ahead control. Control Engineering Practice, 18(11):1318–1327, 2010. HSL. A collection of fortran codes for large scale scientific computation. http://www.hsl.rl.ac.uk, 2013. Adam Lowell Schwartz. Theory and implementation of numerical methods based on Runge-Kutta integration for solving optimal control problems. PhD thesis, University of Berkeley, California, 1996. Martin Sivertsson and Lars Eriksson. Modeling for optimal control: A validated diesel-electric powertrain model. In SIMS 2014 - 55th International Conference on Simulation and Modelling, Aalborg, Denmark, 2014.
A.1 Torque model Mice = Mig − Mf ric − Mpump Vd (pem − pim ) Mpump = 4π Vd 5 2 10 cf r1 ωice + cf r2 ωice + cf r3 Mf ric = 4π uf 10−6 ncyl qHV ηig Mig = 4π 1 ηig = ηig,t (1 − γcyl −1 ) rc ηig,t = Mf,1 + gf (Mf,2 − Mf,1 ) 1 + tanh(0.1(ωice − 1500π/30)) gf = 2 2 Mf,1 = cMf,1 ,1 ωice + cMf,1 ,2 ωice Mf,2 =
2 cMf,2 ,1 ωice
+ cMf,2 ,2 ωice + cMf,2 ,3
A.2 Volumetric efficiency √ √ ηvol = cvol,1 pim + cvol,2 ωice + cvol,3
(A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9)
(A.10)
A.3 Exhaust pressure and temperature qin =
m ˙ f qHV m ˙ f +m ˙ ac
a 1−γa Teo =ηsc Π1−1/γ rc e
(A.11)
qin + Tim rcγa −1 cp,a −
htot Vpipe
Tem =Tamb + (Teo − Tamb )e (m˙ f +m˙ ac )cp,e dpem Re Tem = (m ˙ ac + m ˙ f −m ˙ t−m ˙ wg ) dt Vem 71
(A.12) (A.13) (A.14)
ScienceDirect
IFAC-PapersOnLine 48-15 (2015) 066–071 Model and discretization impact on Model and discretization impact on oscillatory optimal control for aon Model and discretization impact oscillatory optimal control for a diesel-electric powertrain oscillatory optimal control for a diesel-electric powertrain diesel-electric Martin Sivertsson ∗ powertrain Lars Eriksson ∗
PPgen P [kW] [kW] P[kW][kW] gen gen gen
2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 66 Hosting by Elsevier Ltd. All rights reserved. Peer review© of International Federation of Automatic Copyright 2015 IFAC 66 Copyright ©under 2015 responsibility IFAC 66 Control. 10.1016/j.ifacol.2015.10.010
gen gen gen gen
[kW] P[kW]P[kW][kW] PP
ice ice ice ice
ice ice ice ice
ωω [rpm] ω[rpm] ω[rpm] [rpm]
Eriksson ∗ ∗ Eriksson Eriksson ∗ Link¨ Engineering, oping Engineering, Link¨ oping {marsi, larer}@isy.liu.se. Vehicular Systems, Dept. of Electrical Engineering, Link¨ oping ping University, SE-581 83 Link¨ oping, Sweden, Engineering, {marsi, larer}@isy.liu.se. Vehicular Systems, Dept. of Electrical Link¨ o University, o University, SE-581 SE-581 83 83 Link¨ Link¨ oping, ping, Sweden, Sweden, {marsi, {marsi, larer}@isy.liu.se. larer}@isy.liu.se. Abstract: A mean value engine model is used to study optimal control of a diesel-electric Abstract: mean value engine is used studytooptimal control of a diesel-electric powertrain. AThe resulting optimalmodel controls are to shown be highly oscillating for certain Abstract: A mean value engine model is to study optimal control of powertrain. resulting optimal controls arethis shown be highly oscillating formodeling certain Abstract: AThe mean valuethe engine model is used used to study optimal control of aa diesel-electric diesel-electric operating points, raising question whether is antoartifact of discretization, powertrain. resulting optimal controls are shown be oscillating for certain operating raising the question whether is anto of discretization, powertrain. The resulting optimal controls arethis shown toartifact be highly highly oscillating formodeling certain choices or points, a The phenomenon available in real engines. Several model extensions are investigated operating points, raising the question whether this is an artifact of discretization, modeling choices or a phenomenon available in real engines. Several model extensions are investigated operating points, raising the question whether this is an artifact of discretization, modeling and their corresponding optimal control trajectories are studied. It is shown that the oscillating choices or a phenomenon phenomenon available in real real engines. Several model extensions arethe investigated and their corresponding optimal control trajectories are studied. It is shown that oscillating choices or a available in engines. Several model extensions are investigated controls cannot be explained by the implemented extensions to the previously published model, and corresponding optimal control are studied. It shown that the oscillating controls cannot be explained by the implemented extensions to the previously published model, and their corresponding optimal control trajectories are studied. It is is shown that the solution oscillating nor their by the discretization, showing thattrajectories for certain operating points the optimal is controls cannot be by extensions to previously published model, nor by the discretization, that for certain operating points the optimal solution is controls cannot be explained explainedshowing by the the implemented implemented extensions to the the previously published model, periodic. nor by by the the discretization, discretization, showing showing that that for for certain certain operating operating points points the the optimal optimal solution solution is is periodic. nor © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. periodic. periodic. 200 1. INTRODUCTION 2400 200 9 1 2 3 4 5 6 10 8 11 1. INTRODUCTION 7 2400 200 9 4 5 6 10 8 11 1. INTRODUCTION 2000 1 2 3 7 200 1. INTRODUCTION 2400 The engine speed of a conventional vehicle is normally 2400 9 4 10 8 11 2000 1 7 9 1 2 2 3 4 5 5 6 6 10 3 8 11 7 The engine speed of a conventional vehicle is normally 100 decided by the wheel speed and the gear ratio. In a diesel1600 2000 2000 The engine speed of speed conventional vehicle isInnormally normally 100 decided by the wheel and thepath gear ratio.is a dieselThe engine speed of aamechanical conventional vehicle 1600 electric powertrain this between the com1200 100 decided by the wheel wheel speed and is thereplaced gear between ratio. In diesel1600 100 electric mechanical path comdecided by the speed and the gear ratio. aaelectric diesel1600 bustion powertrain engine andthis the wheels by In anthe 1200 electric powertrain this mechanical path between the combustion engine and the wheels is replaced by an electric 800 electric powertrain this mechanical path between the com0 path instead. This introduces an extra degree of freedom 1200 200 400 600 800 1000 1200 1400 1600 1800 1200 bustion engine and the can wheels iscontrolled replaced by of an freedom electric 800 0 time [s] path instead. This introduces extra degree bustion and the wheels replaced by an electric since theengine engine speed beanis independently 200 400 600 800 1000 1200 1400 1600 1800 800 0 time [s] path instead. This introduces ancontrolled extra degree of freedom freedom 0 since the engine speed can offers bean independently path This introduces extra degree of 200 400 600 800 1000 1200 1400 1600 1800 WHTC and its traction phases. of theinstead. wheel speed, which the potential of both Fig.8001. The 200 400 600 800 1000 1200 1400 1600 1800 time [s] [s] since the engine speed can offers be controlled independently 150 traction phases. of thethe wheel which the potential of both since engine speed can be independently optimizing thespeed, performance andcontrolled consumption since the Fig. 1. The WHTC and its time 20001. The is 150 hypothesis that the oscillations seen in the optimal Fig. WHTC and its traction phases. of the wheel speed, which offers the potential of both optimizing the performance and consumption since the Fig. 1. The WHTC and its traction phases. of the wheel speed, which offers the potential of both operating point of the diesel engine can be controlled more 2000 150 hypothesis is that the oscillations seen in the optimal 150 100 optimizing the performance and consumption since the variable geometry turbine (VGT) control of a diesel engine operating the diesel engine can be controlled more optimizing the performance and consumption the freely thanpoint in aof conventional powertrain. This since of course 2000 hypothesis is that the oscillations seen in the optimal 2000 100 is variable geometry turbine (VGT) control of a diesel engine hypothesis is that the oscillations seen in the optimal 1500 are due to decrease in the gas exchange losses. This operating point of the diesel engine can be controlled more freely in aofconventional This degree of course operating theofdiesel be extra controlled more raises than the point question how engine topowertrain. usecan this of variable 100 geometry turbine (VGT) control of aa diesel engine 1500 are due to decrease in the gas exchange losses. This is 100 variable geometry turbine (VGT) control of diesel engine 50 due to that the exhaust manifold pressure oscillates with freely than in a conventional powertrain. This of course raises the question of how to use this extra degree of freely than in a conventional powertrain. This of course freedom. 1500 are due to decrease in the gas exchange losses. This is 50 1500 due to that the exhaust manifold pressure oscillates with are due to decrease in the gas exchange losses. This is raises the the question question of of how how to to use use this this extra extra degree degree of of the1000VGT position whereas the intake manifold pressure freedom. raises 50 due to that the exhaust manifold pressure oscillates with 50 1000 the VGT position whereas the intake manifold pressure 0 Previously it has been studied how to optimally control the due to that the exhaust manifold pressure oscillates with remains unaffected due675to the turbocharger dynamfreedom. 655 660 665 670 680 slower 685 690 695 700 705 freedom. time [s] intake manifold pressure VGT position whereas the Previously has been studied howpower to optimally control the the 1000 remains unaffected due to the slower turbocharger dynampowertrain itbetween two different levels, see Sivertsthe VGT whereas the intake manifold pressure ics.1000 This could that optimal solution is in0 fact 655 660position 665indicate 670 675 680 685 690 695 700 705 time [s] Previously it has been studied how to optimally control the remains unaffected due to the slower turbocharger dynampowertrain between two different power levels, see SivertsPreviously it has been studied how to optimally control the ics. This could indicate that the optimal solution is705 in00 fact son and Eriksson [2015a,b]. For off-highway machinery remains unaffected due675 to slower turbocharger dynam655 660as 665 665 670 675 680 685 690 695 Gilbert 700 705 periodical described inthe Gilbert [1977]. 655 660 670 680 690 695 700 time [s] 685 [1976], time [s] powertrain between two different power levels, see Sivertsics. This could indicate that the optimal solution is in fact fact son and patterns Eriksson [2015a,b]. For off-highway machinery the periodical powertrain between different power levels, see Sivertsas described in Gilbert [1976], Gilbert [1977]. driving aretwo normally very transient, something ics. This could indicate that the optimal solution is in Other possible explanations are either that the solution is son and Erikssonin[2015a,b]. [2015a,b]. For off-highway machinery the Other periodical as described described in that Gilbert [1976], Gilbert [1977]. driving patterns are very transient, something son Eriksson For off-highway machinery the explanations are the either that Gilbert the is thatand is captured thenormally World Harmonized Transient Cycle periodical as Gilbert [1976], [1977]. along apossible singular arc andin controls aresolution therefore driving patterns are normally very transient, something explanations are either that the solution is that is captured inare thenormally World Harmonized Transient Cycle driving patterns very transient, something along apossible singular arc and controls are therefore (WHTC), see WHDC Working Group [2005], shown in Other Other possible explanations are the either that or the solution is oscillatory, as discussed in that Schwartz [1996], that it is an that is captured in the World Harmonized Transient Cycle along a singular arc and that the controls are therefore (WHTC), see WHDC Working Group [2005], shown in that is captured in the World Harmonized Transient Cycle as discussed in that Schwartz [1996], orare that it is an Fig. 1. The WHTC can be divided into 11 traction phases, oscillatory, along a singular arc and controls integration error exploited by the algorithm to therefore decrease (WHTC), see WHDC Working Group [2005], shown in integration oscillatory, as discussed inHellstr¨ Schwartz [1996], or that that it both is an an Fig. 1. The WHTC can be divided into 11[2005], traction phases, (WHTC), see WHDC Working shown in error exploited by the to decrease defined as the period between twoGroup idle periods, where idle oscillatory, Schwartz [1996], or it is the criteriaas asdiscussed shown inin om algorithm et al. [2010]. In Fig. 1. The WHTC can be divided into 11 traction phases, integration error exploited by the the algorithm to decrease decrease defined as the period between two into idle periods, idle the Fig. 1. The WHTC can bethe divided 11 traction criteria as would shown in suspected Hellstr¨ om algorithm et al.the [2010]. In both is assumed to occur when engine speed is the where idlephases, speed integration exploited by to these cases error it be that frequency of defined as the the period between two idle periods, where idle the criteria as shown in Hellstr¨ oom et [2010]. In is assumed to occur when the engine speed is the where idleengine speed defined as between two idle periods, idle these cases it be suspected frequency of and no power isperiod required. Here the phases where the the oscillations criteria as would shown inon Hellstr¨ m that et al. al.the [2010]. In both both depend the discretization. To test this is assumed assumed toi.e. occur when the engine speed is the the the idle speed these cases would be suspected that the of and no power is required. the phases is to occur the speed is idle speed oscillations depend thepoint discretization. To this motored, Pgenwhen < 0 Here are engine ignored andwhere P inengine those the these cases ait itsingle would beon suspected that the frequency frequency of hypothesis operating is selected andtest studied gen is and noset power is required. required. Here the phases phases where theinengine engine the oscillations depend on the discretization. To test this are ignored andwhere Pgen is those hypothesis is motored, Pgen < 0 Here and no power is the the a single operating point is selected and studied cases to i.e. zero. the oscillations depend on the discretization. To test this using a very fine time discretization. is motored, motored, i.e. Pgen hypothesis single operating point is is selected selected and and studied studied gen < gen is cases set to i.e. zero.P <0 0 are are ignored ignored and and P Pgen is in in those those using is a veryaa fine time discretization. hypothesis single operating point To investigate the potential of the diesel-electric power- using a very fine time discretization. cases set to to zero. zero. cases set using a very fine time discretization. To investigate potential of the extra diesel-electric power2. CONTRIBUTIONS train and how the to best exploit degree of freeTo potential of diesel-electric power2. CONTRIBUTIONS train and how the tobybest degree of freeTo investigate the potential of the the extra diesel-electric powerdominvestigate introduced the exploit electrification of the powertrain, 2.ofCONTRIBUTIONS CONTRIBUTIONS train and how to best exploit the extra degree of freedom introduced by the electrification of the powertrain, train and how to best exploit the extra degree of freeminimizing fuel for the WHTC is cast as an optimal The contributions2. this paper is a deeper study of the dom introduced introduced by the electrification of the the powertrain, minimizing fuel by for theelectrification WHTC is cast as powertrain, an optimal The contributions of this paper for is adiesel-electric deeper study powerof the dom the of control problem (OCP). In a conventional powertrain occurence of oscillating controls minimizing fuel the WHTC is an optimal of this paper is aadiesel-electric deeper study of the control problem (OCP). In a speed conventional powertrain occurence oscillating controls minimizing fuel for for the engine WHTC is cast cast as an optimal The contributions of for this paper for iscontrol deeper study powerofMore the WHTC prescribes both and as output power, The trainscontributions as aofsolution optimal problems. control problem (OCP). In a conventional powertrain occurence of oscillating controls for diesel-electric powerWHTC both isengine output power, as aofit solution optimalthe control problems. More control problem (OCP). In a speed conventional powertrain occurence oscillating controls for diesel-electric powerbut hereprescribes engine speed a degree ofand freedom and also trains specifically studiesforwhether observed oscillations WHTC prescribes bothisisengine engine speed and output power, trains as solution for optimalthe control problems. More but hereprescribes engine a degree ofand freedom also specifically studies whether observed oscillations WHTC both speed power, trains aa it solution optimal control problems. More optimized. If this speed OCP solved for phase 8 output in theand WHTC are an as artifact of thefor discretization. It also investigates if but resulting here engine engine speed issolved a very degree of freedom freedom and also specifically it studies studies whether the observed oscillations optimized. If this OCP isare foroscillatory, phase 8 in see theand WHTC an artifact of thebediscretization. It investigates if but here speed is a degree of specifically it whether oscillations the controls Fig.also 2, are the oscillations can explainedthe by observed thealso models used and optimized. If OCP for phase in the an artifact of the It investigates if the resulting controls Fig.and 2, are the oscillations can explained byimpacts thealso models used and optimized. If this this OCP isare solved foroscillatory, phasein8 8Sivertsson in see the WHTC WHTC are an artifact of thebediscretization. discretization. It also investigates if t ∈ [670, 678], [684, 687].is Itsolved isvery mentioned whether or not extending the model the oscillating the resulting controls are very oscillatory, see Fig. 2, the oscillations can be explained by the models used and tEriksson ∈ [670, 678], [684, 687]. It is mentioned in Sivertsson and whether or not extending the model impacts the oscillating the resulting controls are optimal very oscillatory, seetransient Fig. 2, the oscillations can be also explained by the models and [2015a] that the solutions in solutions. The paper presents a fast and used accurate ∈ [670, [670, control 678], [684, 687]. It is is mentioned in Sivertsson Sivertsson and solutions. whether or not extending thepresents model impacts the oscillating Eriksson [2015a] the optimal solutions in are transient The paper also fastoptimal andoscillating accurate tt ∈ 678], [684, 687]. It mentioned in and whether not extending the model the optimal ofthat a diesel-electric powertrain often residual or gas model suitable for use impacts ina an control Eriksson control [2015a] that the optimal optimal solutions in are transient solutions.gas The paper also presents presents fastoptimal and accurate accurate optimal a diesel-electric powertrain often residual model suitable for use inaa an control Eriksson [2015a] the solutions in transient solutions. The paper also fast and oscillatory and inofthat Asprion et al. [2014] the unconfirmed context. optimal aa diesel-electric powertrain are oscillatory and inof et al. [2014] the unconfirmed context. optimal control control of Asprion diesel-electric powertrain are often often residual residual gas gas model model suitable suitable for for use use in in an an optimal optimal control control oscillatory and in Asprion et al. [2014] the unconfirmed context. oscillatory and in Asprion et al. [2014] the unconfirmed context. Copyright © 2015 IFAC 66 ωω [rpm] ω[rpm] ω[rpm] [rpm]
Martin Sivertsson ∗ Lars ∗ ∗ Lars Martin ∗ Martin Sivertsson Sivertsson Lars Vehicular Systems, Dept. of Electrical ∗ Vehicular Systems, Dept. of Electrical University, SE-581 83 Link¨ o ping, Sweden, ∗ ∗
pim [kPa]
ωice [rad/s]
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180 160 140 120 100 80 655
pem [kPa] ωtc [rad/s] f
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u [mg/cycle]
[−] wg
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120 100 80 60 40 20
660
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150 100 50 0 655
time [s]
Fig. 2. The optimal solution to phase 8 of the WHTC, with ωice as a degree of freedom. The resulting controls are highly oscillatory, see t ∈ [670, 678], [684, 687] Description Pressure Temperature Rotational speed Massflow Power Torque Energy Pressure ratio Volume Specific heat capacity ratio Specific heat capacity constant pressure Specific heat capacity constant volume Gas Constant Control signals Inertia Blade speed ratio Fuel-air equivalence ratio Air-fuel smoke-limit Residual gas fraction Fuel to mass ratio Lower heating value Compression ratio
Index GenSet em ac wg tc e mech c, surge d pump eo
Description Engine-Generator Exhaust manifold After compressor Wastegate Turbocharger Exhaust Generator-mechanical Compressor surge-limit Displaced Pumping Engine out
4. PROBLEM FORMULATION
Table 1. Symbols used Symbol p T ω m ˙ P M E Π V γ cp cv R uf , uwg , Pgen J BSR φ λmin xr MF R qHV rc
Description Engine Intake manifold Compressor Turbine Fuel Air Generator-electrical Reference Volumetric Friction Indicated gross Ambient
turbocharger speeds, ωice/tc , and inlet and exhaust manifold pressures, pim/im . The controls are injected fuel mass, uf , wastegate position, uwg , and generator power, Pgen . The engine model consists of two control volumes, intake and exhaust manifold, and four restrictions, compressor, engine, turbine, and wastegate. The governing differential equations of the MVEM are: dωice Pice − Pmech = (1) dt ωice JGenSet dpim Ra Tim = (m ˙ c−m ˙ ac ) (2) dt Vim dpem Re Tem = (m ˙ ac + m ˙ f −m ˙ t−m ˙ wg ) (3) dt Vem dωtc Pt ηtm − Pc = (4) dt ωtc Jtc For a complete list of the symbols used in the paper, see Table 1-2.
1 0.8 0.6 0.4 0.2 655
[kW]
665
150
10000 8000 6000 4000 2000 655
gen
660
Index ice im c t f a gen ref vol f ric ig amb
200
655
P
Table 2. Subscripts used
250
100 655
67
This paper uses the MVEM to study optimal stationary operation, or lack of it in the case of oscillating controls.
Unit Pa K rad/s kg/s W Nm J m3 J/(kg · K) J/(kg · K) J/(kg · K) mg/cycle, -, W kg · m2 J/kg -
4.1 Stationary optimization As a reference for the dynamic optimization, three stationary optimization problems are first solved, to find the following three stationary points for the given ωref , Pref combination: The maximum efficiency, φmax , the maximum fuel/air-ratio, ηmax , and the minimum fuel/air-ratio, P φmin . η = m˙ fgen qHV is the efficiency of the powertrain and φ is the fuel/air-ratio. These problems are solved to find the optimal operating point for stationary operation and also the limits for stationary operation. 4.2 Dynamic optimization The main optimal control problem studied is: T m ˙f min
3. MODEL
u(t)
0 (5) x(t) ˙ = f (x(t), u(t)) (x(t), u(t)) ∈ Ω(t) where x is the state vector of the MVEM, x˙ is the state equations (1)-(4), and u = [uf , uwg , Pgen ]. The optimal control problems are also subject to a set of constraints, namely:
The basic model used can be downloaded in the LiU-DEl-package from systems software [2014] and is described in detail as M V EMo in Sivertsson and Eriksson [2014]. The modeled diesel-electric powertrain consists of a 6cylinder diesel engine with a fixed-geometry turbine and a wastegate for boost control, with a generator mounted on the output shaft. The states of the MVEM are engine and
s.t.
67
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ωice [rad/s]
x(0) = x(T ) = x(ηmax ), x(T ˙ )=0 xmin ≤ x(t) ≤ xmax umin ≤ u(t) ≤ umax , ωice (T ) = ωref or ωice (t) = ωref , Pgen (t) = Pref 1 Pice (x(t), u(t)) ≤ Pice,max (x(t)), φ(x(t), u(t)) ≤ λmin BSRmin ≤ BSR(x(t), u(t)) ≤ BSRmax , Πc ≤ Πc,surge (6) The constraints are actuator and state limits, as well as constraints imposed by the components, such as maximum power of the engine, Pice , surge-limit of the compressor, Πc,surge , blade speed ratio-limit of the turbine, BSR, as well as environmental constraints, i.e. an upper limit on φ set by the smoke-limiter.
185 180 175 170 0
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1
0
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0.9
1
0 4 x 10
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1
0
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1
0
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0.9
1
pem [kPa]
pim [kPa]
300 250 200
250 200
ωtc [rad/s]
150
uf [mg/cycle]
The driving mission-constraints are that the powertrain starts in the operating point of maximum efficiency ηmax , a point it should also end in, with the added requirement that the end operating point should be stationary. The generator power is also fixed to the reference value. Two types of problems are then studied, one where the engine speed is fixed to the reference speed, denoted ωice = fix, and one where it is allowed to depart from this as long as it starts and ends in ωice = ωref , denoted ωice = free.
1.2 1 0.8
120 100 80
uwg [−]
1 0.5 0
Pgen [kW]
5. NUMERICAL SOLUTION The software package that is used to solve the optimal control problem numerically is CasADi Andersson [2013]. First the problem is discretized using Radau collocation with three collocation points in each control interval. The states are thus approximated with a third order polynomial, whereas the controls are approximated by a second order polynomial in each control interval. The states are required to be continuous over each control interval boundary, whereas the controls are allowed to be discontinuous. The resulting nonlinear program(NLP) is solved using IPOPT, W¨ achter and Biegler [2006], with the MA57 linear solver from the HSL package, HSL [2013]. For the wastegate oscillation study 200 control intervals have been used.
145 140 135 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x(ηm ax) x(φm ax) x(φm in) ωice = fix ωice0.9 = free
1
time [s]
Fig. 3. Optimal states and controls for constant output power, both stationary and dynamic. The dynamic solutions are highly oscillatory. losses only depend on engine speed except the pumping torque, Mpump , see Appendix A.1. This means that the oscillating control actually decreases the pumping torque, as hypothesised in Asprion et al. [2014]. Looking at Fig. 3 the low pass filtering effect of the turbocharger can be clearly seen since the wastegate opening and closing results in a pem span of 60-100 kPa depending on if ωice = free/fix, whereas the effect on pim is only 4-5 kPa. In Fig. 4 the pumping power, Ppump = ωice Mpump , are shown relative Ppump (ηmax ). Both x(φmin ) and x(φmax ) increase Ppump compared to x(ηmax ), as expected. The oscillation changes Ppump several hundred percent away from what is stationary optimal.
6. OSCILLATING CONTROLS The stationary point of interest here is the one seen with oscillating controls in Fig. 2 namely Pgen = 140 kW and ωice = 1700 rpm. The three stationary OCPs (OSS) described in Section 4.1 and the two dynamic OCPs (OSD), ωice = fix and ωice = free, described in Section 4.2 are solved using CasADi/IPOPT/HSL and the results are shown in Fig. 3. Looking at Fig. 3 it is clearly visible that both OSDs result in a periodic oscillation. If the engine speed is free the amplitude and frequency changes slightly, however the nature of the oscillation remains the same. The nature of the opening and closing goes against the hypothesis that this should be oscillations due to a singular arc or an effect of the integration error since the wastegate only opens 9-11 times despite 200 control intervals with a three controls in each interval, yielding an optimal period of 90-110ms which is approximately 20 times the control interval length. This indicates that the oscillations are in fact optimal.
In Table 3 the gains of oscillating controls are quantified. x(φmax ) and x(φmin ) both increase the pumping energy Epump with 50% which also leads to a relative efficiency decrease of 0.5% (0.2% absolute). The oscillating control with fixed ωice decreases Epump with 2.4% and ωice free with 4.1%. Since the friction losses are quadratic in engine speed, ωice -free, increases the friction losses, but it is still beneficiary since the relative efficiency increase is 0.52 vs. 0.24 for ωice -fix (0.2 vs. 0.09 absolute). The gains are small but nevertheless surprising since it’s a dynamic phenomenon. Looking at Fig. 5 the wastegate’s effect on the stationary efficiency as well as pumping torque is shown. The efficiency is a convex function in uwg whereas Mpump is concave, which if the analysis was per-
This is especially interesting for the case with fixed engine speed, since then the effect can be isolated since all torque 68
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satisfies both conservation of mass and energy is implemented.
300
x(φm ax) x(φm in) ωice = fix ωice = free
Ppump, rel [%]
200 100
The basic model is M V EMo which is then extended with different combinations of these three models. The model fit for the different models versus both dynamic(Dyn.) and stationary measurements(Stat) are shown in Table 4. None of the model extensions have any significant effects on the model fit versus measurements.
0 −100 −200 −300 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time [s]
Fig. 4. Pumping power relative the stationary optimal point. x(φmax ) x(φmin ) ωice = fix ωice = free
∆η -0.542 -0.482 0.024 0.052
∆Ef ric 0.0 0.0 0.0 0.8797
∆Epump 56.3782 50.1279 -2.4073 -4.0869
7.1 Residual gas In Wahlstr¨om and Eriksson [2011] the engine out temperature model is based on an ideal Seiliger cycle model, incorporating residual gas. The model is formulated as: xr,0 = 0,Teo,0 = 800. While Teo,k+1 − Teo,k >1e-6 m ˙ f qHV qin = (1 − xr,k ) m ˙ f +m ˙ ac qin xcv xp =1 + cv,a T1 rcγa −1 qin (1 − xcv ) xv =1 + qin xcv cp,a ( cv,a + T1 rcγa −1 ) (7) 1/γ −1/γ Πe a xp a xr,k+1 = rc x v T1 =xr,k+1 Teo,k + (1 − xr,k+1 )Tim
∆Eig 0.5448 0.4844 -0.0223 -0.0263
0.3735
−5
pump
[Nm]
0
M
η [−]
Table 3. Changes in efficiency and energies T relative x(ηmax ) in percent, Ex = 0 Px dt. 0.375
0.372
0
0.1
0.2
0.3
0.4
0.5
u
wg
0.6
0.7
0.8
0.9
1
69
−10
[−]
a 1−γa 1/γa −1 Teo,k+1 =ηsc Π1−1/γ rc xp e xcv 1 − xcv γa −1 + qin + T 1 rc cp,a cv,a The equations in (7) are nonlinear and depend on each other and need to be solved using fixed point iterations. In Wahlstr¨om and Eriksson [2011] it is shown that if the solution from the previous time step is known, one iteration suffices to get a good approximation of the engine out temperature. In an optimization context it is difficult to keep track of the solution form the previous time step since the time steps are solved simultaneously. Also a submodel that is evaluated for a varying number of iterations is undesirable, especially since it complicates the computation of derivatives. Therefore this type of model is not implementable in an optimization context and a new model is developed.
Fig. 5. Wastegate position’s effect on the efficiency and pumping torque during stationary conditions. formed for stationary conditions would lead to the conclusion that oscillating controls would increase the pumping torque and consequently decrease the efficiency. However the result is actually the opposite, oscillating between the two worst controls from a stationary perspective, increases the efficiency dynamically. 7. MODEL EXTENSIONS To investigate whether the oscillating controls are results of a modeling assumption a set of different model extensions are considered. The extensions and their motivation are: • xr : Model for residual gas: In Sivertsson and Eriksson [2014] it is shown that mean/max absolute relative error increase of assuming xr = 0 is [0.014/0.06] versus measurements. However in the oscillating controls the exhaust pressure changes very rapidly, something that might have a significant effect on the amount of residual gas trapped in the cylinder. • ηvol,2 : Pressure ratio dependent volumetric efficiency model. In the volumetric efficiency model used, see Appendix A.2, only the dependence of the intake manifold pressure and engine speed are modeled. Of course the changing pressure ratio over the engine could have effects on the volumetric efficiency. • Adiabatic: The isothermal exhaust manifold model used, see Appendix A.3, neglects energy conservation since it assumes that the gases flowing in and out of the control volume have the same temperature, therefore an adiabatic exhaust manifold model that
xr is modeled as a function of both pressure ratio over the engine and the fuel to mass ratio. defined as: m ˙f (8) MFR = m ˙ f +m ˙ ac xr,mod1 cx Π2 + cxr2 Πe + cxr3 (9) = r1 e xr = xr,mod2 1 + cxr4 M F R In Fig. 6-top this gives a good agreement to the iterative model, resulting in mean/max relative errors of [0.92/4.52] %. Using that xcv becomes zero in the optimization for this particular engine Teo can be computed according to Teo =
MF R + Tim rcγa −1 ) (1 − xr )( qHVcp,a 1
1−1/γa 1−γa rc
ηsc Πe
− xr rcγa −1
(10)
Teo and xr could be used as starting values and then one iteration of the fixed point iteration can be performed. However it turns out that not only is it more computations, 69
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0.14
xr /xr,m od1 [−]
1.4
0.1 0.08 0.06 0.04 0.02
1.2 1 0.8
Dyn. M V EMo xr xr +adiabatic ηvol,2 ηvol,2 + xr +adiabatic Stat. M V EMo xr xr +adiabatic ηvol,2 ηvol,2 + xr +adiabatic
0.6 0.4
0
0.2 0
1
2
3
4
0
0.02
Πe [−]
0.04
0.06
0.08
0.1
M F R [−] T :x
1200
eo
r, iter
T :x eo
r, mod
ωice T V 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 m ˙c Pc 2.5 1.8 2.5 1.8 2.5 1.8 2.5 2.0 2.5 1.9
pim T V 2.8 2.2 2.8 2.2 2.8 2.2 2.8 2.3 2.9 2.3 m ˙ ac Tem 2.5 2.4 2.4 2.5 2.5 2.7 2.7 2.3 2.7 2.6
pem T 2.8 2.8 2.8 2.9 2.9 m ˙ exh 3.3 3.3 3.1 3.2 3.0
ωtc V 2.9 2.9 3.0 3.0 3.0 Pt 5.4 5.5 4.9 5.5 4.9
T 2.9 2.9 2.9 3.0 3.1
V 2.9 2.9 3.0 3.2 3.2
+ Pmech 3.4 3.3 3.1 5.1 4.4
− Pmech 1.4 1.5 1.5 1.4 1.6
800
Ppump, rel [%]
MVEM 600 400 660
665
670
675
680
685
690
695
700
705
Ppump, rel [%]
655
rel error [%]
Ppump, rel [%]
Fig.0.056. Top: Residual gas, xr , vs. pressure ratio, Πe , and how this dependence is modeled (Left). The xr error’s dependence on M F R (Right). Bottom: Fit of the new 0 residual gas model for phase 8 of the WHTC. the−0.05 model fit is actually worse. When simulated over the entire WHTC the mean/max absolute relative error of −0.1 exhaust gas temperature, Teo for the model in (8)-(10) 655 660 665 675 680 685 690 700 705 relative the model in670(7) are [0.025/0.22]%. If 695 in addition to time [s] (8)-(10) one fixed point iteration is used the errors increase to [0.045/0.34] %. The model used is therefore the one without iterations. In Fig. 6-bottom the fit vs. the model in (7) is shown.
o
400 200 0 −200 −400
x
r
400 200 0 −200 −400
ηvol, 2
400 200 0 −200 −400
Ppump, rel [%]
Teo [K]
1000
400 200 0 −200 −400 −600
Ppump, rel [%]
xr [−]
Table 4. Mean absolute relative errors for the different models versus measurements. T=tuning set, V=validation set.
1.6
xr xr,m od1
0.12
400 200 0 −200 −400 −600
xr+adiabatic
η
+x +adiabatic
vol, 2
r
ωice=fix ωice=free 0
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0.9
1
Fig. 7. Relative pumping power of the optimal solutions to the two OSDs for the different models, relative the stationary optimal operating point. The oscillatory solution can not be explained by any of the model extensions.
7.2 ηvol,2 : Modified volumetric efficiency The ηvol,2 model implemented is a modified version of the model found in Eriksson and Nielsen [2014], consisting of an ideal part and two polynomials, in ωice and pim respectively. 1/γa rc − ppem im ηvol,ideal = rc − 1 2 ηvol,ωice =cηvol,1 ωice + cηvol,2 ωice + cηvol,3 ηvol =ηvol,ωice ηvol,ideal + cηvol,4 pim + cηvol,5 The component model fit for stationary measurements are slightly better than the model in (A.10), with mean/max absolute relative errors of 0.62/2.68 % vs. 0.9/3.7 % for (A.10), which is to be expected since the number of tuning parameters increases from three to five.
the manifold is computed according to (A.13), that is temperature after heat loss. 8. RESULTS The two OSDs defined in Section 4.2 are solved for the different models and the resulting optimal pumping power, Ppump is plotted relative the stationary optimal in Fig. 7. The oscillations cannot be explained by any of the model extensions. The periodic nature of the solution is present for all model extensions and the changes in frequency and amplitude of the oscillations are minor. In Table 5 the changes in energies and efficiency are shown, which confirms the results seen in Fig. 7. The decrease in pumping energy, ∆Epump , increases for each model extension for ωice = fix, indicating that given a standard mean value engine model it actually is optimal to use periodic wastegate control in order to decrease the pumping torque.
7.3 Adiabatic exhaust manifold model
The adiabatic model, as described in Chevalier et al. [2000] is implemented according to: dpem Re γe = (Tem,k (m ˙ ac + m ˙ f ) − Tem (m ˙ t+m ˙ wg )) dt Vem 9. CONCLUSIONS Re Tem dTem = ˙ ac + m ˙ f ) − Tem (m ˙ t+m ˙ wg ) γe Tem,k (m dt pem Vem Optimal control of a diesel-electric powertrain is studied. It is shown that the oscillatory solutions seen for certain − Tem (m ˙ ac + m ˙ f −m ˙ t−m ˙ wg ) operating points are not directly discretization dependent. This means extending the model with an additonal state, Instead the solution is periodic with a period much greater Tem . Tem,k , i.e. the temperature of the gases flowing into than the control interval length. Further it is seen that 70
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Martin Sivertsson and Lars Eriksson. Optimal transient control trajectories in diesel-electric systems-part 1: Modeling, problem formulation and engine properties. Journal of Engineering for Gas Turbines and Power, 137(2), February 2015a. Martin Sivertsson and Lars Eriksson. Optimal transient control trajectories in diesel-electric systems-part 2: Generator and energy storage effects. Journal of Engineering for Gas Turbines and Power, 137(2), February 2015b. Vehicular systems software. ”http://www.fs.isy.liu. se/Software/”, 2014. Andreas W¨achter and Lorenz T. Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 106:25–57, 2006. Johan Wahlstr¨om and Lars Eriksson. Modelling diesel engines with a variable-geometry turbocharger and exhaust gas recirculation by optimization of model parameters for capturing non-linear system dynamics. Proceedings of the Institution of Mechanical Engineers, Part D, Journal of Automobile Engineering, 225(7):960–986, 2011. WHDC Working Group. Worldwide harmonized heavy duty emissions certification procedure. ”http:// www.unece.org/fileadmin/DAM/trans/doc/2005/ wp29grpe/TRANS-WP29-GRPE-50-inf04r1e.pdf”, 2005. GRPE-50-4-Rev.1, Read 2/2-2015.
Table 5. Changes in efficiency and energies of the two OSDs, ωice = fix and ωice = free, relative x(ηmax ), for the different models. M V EMo xr xr + adiabatic ηvol,2 ηvol,2 + xr + adiabatic
fix free fix free fix free fix free fix free
∆η 0.024 0.052 0.024 0.054 0.024 0.068 0.029 0.070 0.030 0.080
∆Ef ric 0.0 0.88 0.0 1.32 0.0 1.74 0.0 -2.23 0.0 -1.99
∆Epump -2.41 -4.09 -2.52 -3.64 -2.58 -3.7 -3.14 -10.16 -3.23 -10.21
71
∆Eig -0.02 -0.03 -0.02 0.02 -0.02 0.19 -0.03 -0.31 -0.03 -0.22
the pumping work of the engine decreases as a result of the oscillations. To study if this effect is a result of oversimplification in the previously published mean value engine model, several model extensions are investigated and their corresponding optimal control problems are solved. Furthermore a new residual gas model, suitable for optimal control, is presented. It is shown that the oscillating controls cannot be explained by the implemented extensions to the previously published model, showing that for certain operating points for mean value engine models the optimal solution is actually periodic. REFERENCES
Appendix A. EXCERPTS FROM ORIGINAL MODEL
Joel Andersson. A General-Purpose Software Framework for Dynamic Optimization. PhD thesis, Arenberg Doctoral School, KU Leuven, Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center, Kasteelpark Arenberg 10, 3001-Heverlee, Belgium, October 2013. Jonas Asprion, Oscar Chinellato, and Lino Guzzella. Optimal control of diesel engines: Numerical methods, applications, and experimental validation. Mathematical Problems in Engineering, 2014. Alain Chevalier, Martin M¨ uller, and Elbert Hendricks. On the validity of mean value engine models during transient operation. In SAE Technical Paper 2000-011261. SAE International, 2000. Lars Eriksson and Lars Nielsen. Modeling and Control of Engines and Drivelines. John Wiley & Sons, 2014. Elmer G. Gilbert. Vehicle cruise: Improved fuel economy by periodic control. Automatica, 12:159–166, 1976. Elmer G. Gilbert. Optimal periodic control: A general theory of necessary conditions. SIAM Journal of Control and Optimization, 15(5):717–746, 1977. Erik Hellstr¨ om, Jan ˚ Aslund, and Lars Nielsen. Design of an efficient algorithm for fuel-optimal look-ahead control. Control Engineering Practice, 18(11):1318–1327, 2010. HSL. A collection of fortran codes for large scale scientific computation. http://www.hsl.rl.ac.uk, 2013. Adam Lowell Schwartz. Theory and implementation of numerical methods based on Runge-Kutta integration for solving optimal control problems. PhD thesis, University of Berkeley, California, 1996. Martin Sivertsson and Lars Eriksson. Modeling for optimal control: A validated diesel-electric powertrain model. In SIMS 2014 - 55th International Conference on Simulation and Modelling, Aalborg, Denmark, 2014.
A.1 Torque model Mice = Mig − Mf ric − Mpump Vd (pem − pim ) Mpump = 4π Vd 5 2 10 cf r1 ωice + cf r2 ωice + cf r3 Mf ric = 4π uf 10−6 ncyl qHV ηig Mig = 4π 1 ηig = ηig,t (1 − γcyl −1 ) rc ηig,t = Mf,1 + gf (Mf,2 − Mf,1 ) 1 + tanh(0.1(ωice − 1500π/30)) gf = 2 2 Mf,1 = cMf,1 ,1 ωice + cMf,1 ,2 ωice Mf,2 =
2 cMf,2 ,1 ωice
+ cMf,2 ,2 ωice + cMf,2 ,3
A.2 Volumetric efficiency √ √ ηvol = cvol,1 pim + cvol,2 ωice + cvol,3
(A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9)
(A.10)
A.3 Exhaust pressure and temperature qin =
m ˙ f qHV m ˙ f +m ˙ ac
a 1−γa Teo =ηsc Π1−1/γ rc e
(A.11)
qin + Tim rcγa −1 cp,a −
htot Vpipe
Tem =Tamb + (Teo − Tamb )e (m˙ f +m˙ ac )cp,e dpem Re Tem = (m ˙ ac + m ˙ f −m ˙ t−m ˙ wg ) dt Vem 71
(A.12) (A.13) (A.14)