Model Predictive Control Based Energy Management Algorithm for a Hybrid Excavator

Transcription

1 ISSN ISRN LUTFD/TFRT SE Model redictive Control Based Energy Management Algorithm for a Hybrid Ecavator atri Thring Department of Atomatic Control Lnd University October 8

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3 Lnd University Department of Atomatic Control Bo 8 SE- Lnd Sweden Athors atri Thring Docment name MASTER THESIS Date of isse October 8 Docment Nmber ISRN LUTFD/TFRT SE Spervisor Johan Åesson Atomatic Control, Lnd Anders Rantzer Atomatic Control, Lnd Eaminer Sponsoring organization Title and sbtitle Model redictive Control Based Energy Management Algorithm for a Hybrid Ecavator Model redictive Control baserad energiontrollalgoritm för en hybrid grävmasin Abstract The Volvo Grop has an ambitios plan to efficiencies its constrction eqipment vehicles with an introdction of a hybrid engine. This Master thesis is based on, as a first approach, to control the energy split in a hybrid constrction eqipment vehicle HCEV with the control strategy Model redictive Control MC. The wor is divided into two main parts, where the first one is to create a piece-wise affine system WA of the nonlinear plant and secondly to define the control strategy. Mltiple linear models are created from the different mathematical descriptions with the Taylor epansion and then divided in space with polytopes to represent the hybrid dynamics properly. After that, the MC strategy is to be created and this is done by defining a cost fnction, where selected variables shold follow some trajectory. At each sample, the MC controller shold compte an optimal control signal dring the prediction and control horizon to minimize the fel consmption, mae the internal combstion engine rn at a favorable rotational speed and save the drability on the battery. The control problem is simlated in Simlin with a pre-defined driving cycle, as can be seen as an artificial ser for the HCEV, which generates a power demand that needs to be satisfied. Keywords Classification system and/or inde terms if any Spplementary bibliographical information ISSN and ey title Langage English Secrity classification Nmber of pages 89 Recipient s notes ISBN

4 . CONTENTS. CONTENTS.... SUMMARY INTRODUCTION Bacgrond Hybrid Constrction Eqipment Vehicles Literatre srvey Motivation to the wor Control goals THEORY AND TOOLS iece-wise affine system Linearization olytope Model redictive Control Simple eample Hard constraints Soft constraints Model redictive Control for iece-wise affine system Matlab and Mlti-arametric toolbo Matlab and Simlin Mlti-arametric toolbo Maple MODELING Model description Mathematical description of the plant Evalation of mappings Linearization of system Linearization points Creating iece-wise affine system Sampling time Handling distrbances Model validation The linearization of the switches ressre difference ower otpt from generator ower otpt from Linearization of the plant WA WA WA MC SYNTHESIS Control objectives... 48

5 6.. Cost fnction Introdction of new control signals Approach : Minimize the fel consmption Approach : Minimize the losses in the system Constraints hysical constraints in the system Tning Eperienced problems MT limitations Comptational time RESULTS MC operating distrbance constant in pred. horizon MC MC MC MC operating on the nonlinear plant CONCLUSIONS REFERENCES References AENDIX mps Engine system Electrical system Generator Swing system Mapping definitions Graphical illstration of maps Switch Constraints Simlin file on the linearization of the switches Simlin file for the MC control Coefficients in the jacobian matrices

6 . SUMMARY The goal is to control the power split in a Hybrid Constrction Machine HCM with the control strategy; Model redictive Control MC. This is done by first creating linear models from the mathematical descriptions of the dynamical behavior of a HCEV. These models are then divided in space to form a iece-wise affine system as an approimation of a hybrid plant. After the MC model has been created, the control strategy is to be defined and this is done with a cost fnction which shall penalize deviations from a set of reference signals, i.e. have the rotational speed be in a specific interval and minimize the fel consmption. The optimization is applied dring the prediction and control horizon in the controller, which introdce the predictive featre in the system. Also, with the se of a MC controller, physical constraints can be taen into accont to simlate the HCM according to reality. The developed controller is then sed in Simlin along with a driving cycle, which generates a power demand from the ecavator, to evalate the controller s performance. Three models with different approimation levels are evalated against each other with a simple I controller operating to choose the most appropriate one. The best one is then sed as a WA model in the MC controller, where three different MC control strategies are simlated. These control strategies shold be optimized in aspects of low fel consmption and other dynamical variables. The reslts from these control strategies show that the control problem is of a very comple strctre and de to limitations on the sed toolbo in Matlab, the Mlti-parametric toolbo, only an online MC controller cold be created. This lead to a high comptational time with a comple controller, i.e. long prediction and control horizon, many linear models in the MC controller and a short sample interval. Only a short prediction horizon cold be sed and therefore, the reslts obtained in this Master thesis are not completely satisfying. For eample, indications have been shown that the fel consmption can be minimized frther with a more intelligent controller, i.e. longer prediction horizon.

7 3. INTRODUCTION 3.. Bacgrond As with the increasing consmption of petrolem fel and with the mch stricter carbon dioide C policy from governments, greater efforts have been applied to replace or transportation vehicles and hence minimize the fel consmption. When it comes to hybrid constrction eqipment vehicles HCEV, a significant redction in fel consmption can be achieved compared to a traditional vehicle de to the fact that the power demand can be divided between both an engine and a battery. Good preferences for a hybrid constrction machine; The option of trning on/off the internal combstion engine. More favorable woring points Regenerative power in the system With the on-going technical development in the hybrid vehicle section, mch attention is given to improving the energy control algorithm ECA to optimize the energy flow and hence receive lower fel consmption. One of the Volvo grop s biggest competitors, Komats, has developed a HCEV that minimizes fel consmption with -4 % [9], bt there are epectations in which the fel consmption can be redced frther with a more intelligent ECA. One of these ftre thoghts on optimization control strategies is Model redictive Control which is an intelligent and comple control algorithm. In Table the ey words in this wor are highlighted. Shortage of epression Meaning ECA Energy control algorithm EGM Engine mode EM Electrical motor EMA Energy management algorithm Energy storage system HCEV Hybrid constrction electrical vehicle ICE Internal combstion engine MC Model redictive Control MT Mlti-parametric toolbo SoE State of energy SWEM Swing electrical machine Table. Clarifications on shortage of epressions 3... Hybrid constrction eqipment vehicles The Volvo grop has an ambitios plan regarding development of HCEVs. One important challenge when developing sch vehicles is to control the energy flows. The control of a hybrid vehicle is more comple compared to a conventional one and the reason is that the power demand is split between a primary power nit normally an engine and a secondary power nit typically a battery. There are qite a lot of differences between a hybrid car and a hybrid constrction eqipment machine, sch as the power demand comes from different actators, namely the hydralic pmps and the swing. The actators are driven throgh hydralic and inclde the boom cylinder, arm cylinder and bcet cylinder in Figre below. Frthermore, the swing is a rotation mechanism that is sitated above the ndercarriage and maes the pper strctre rotates 36 degrees, which also can be seen in Figre. This mechanism is electrically driven. 5

8 boom bom boom cylinder bomcylinder arm cylinder saftcylinder bcet sopa arm saft rotating mechanism Figre. bcet cylinder sopcylinder Basic constrction of a typical hydralic ecavator The hydralic actators are driven with mechanical power and the swing gets its power demand from an EM that is monted on the swing. These actators receive their power demand directly from the ICE or indirectly from the. This is done throgh a generator, which converts the electrical power to mechanical power. Frthermore, the rotating mechanism in Figre is denoted SWEM and receives its electrical power demand directly from the or indirectly from the generator. In the last case mentioned, the mechanical power is transformed to electrical power needed for the EM. The SWEM can also regenerate energy bac to the battery or to the hydralic actators throgh the generator. This is done when the swing rotation mechanism braes and converts the inetic energy to electrical energy. Another difference between an ordinary hybrid vehicle and a HCEV is that the HCEV has several woring modes which correspond to different dynamics and different constraints in the system. Frthermore, changes between high and low power is more freqent for an ecavator than for a traditional car, which leads to more difficlties in controlling the dynamics in the system. In Figre a system overview is shown that illstrates the different power demanding nits and its sppliers.

9 GEN mp hydr pmps Hydralic actators el GEN el SWEM E SWEM AUX E AUX SWEM AUX Figre. System overview, inclding power electronics E, SWEM stands for swing electric machine and GEN for generator. The system contains two sorce of power sppliers, the internal combstion engine in Figre and the energy storage system in Figre. These two shold satisfy the power demanding nits, which are the hydralic actators denoted pmps, the swing electrical machine denoted SWEM and the power to the AUX denoted AUX. In order to convert the mechanical power from the ICE to electrical power or the opposite, a generator is needed. The generator is monted directly on the engine as can be seen in Figre and this assmption is made throghot this Master thesis, i.e. the ICE has the same rotational speed as the generator. As also can be seen in the figre above is the bo denoted Hydralic actators, which inclde the boom, arm and bcet cylinders. The rotational mechanism can be seen on the right side in Figre, above the SWEM bo. The ser interaction in the ecavator is replaced with a driving cycle which generates for each wor mode a power demand from the hydralic pmps and from the swing system in a given time interval. A wor mode is a predefined session of actions, for eample one wor mode is called dig and dmp, which digs one scoop and then rotates 9 degrees and then dmps the diggings. For this mentioned session, a mechanical and an electrical power demand are generated in time. The driving cycle is described frther in Section Literatre srvey From the literatre stdy, it seems that there has been some development regarding hybrid control algorithm for systems similar to this, bt not anything where the control algorithm was base on Model redictive Control MC. The whole description of the ecavator dynamics and simlation reslts from a nmber of woring modes where handed ot from the beginning of the Thesis wor. A lot of the basic wor was omitted from that literatre bt soon after that mch more attention was redirected to theory on Model redictive Control. Some of the most important are displayed below. In the MCtools. Reference Manal a simple approach on the MC theory can be fond and some easy eamples. This is jst a basic theory grond for this Thesis becase this toolbo is not developed for hybrid systems. The manal can be fond in []. 7

10 Most of the theory behind MC is fond in redictive Control with constraints [4], which describe different areas of MC, for eample setting p the state-space form and solving predictive control problems. The most important theories on MC and an easy eample are highlighted in Section 4 in this wor. The Mlti-arametric Toolbo is a Matlab program that is developed to describe and simlate a general hybrid system. In this toolbo, there are possibilities to inclde switches in the system a more detailed eplanation on this can be fond in Section 4, which is important for simlation of HCEV. Theory and implementation of a system with sch characteristics are docmented, with system strctres as linear time-invariant system LTI and what is more interesting in this thesis, iecewise Affine system WA. Frthermore, description on how to define different parameters thorogh in the reglator can also be fond in this toolbo, Mlti-arametric Toolbo MT [7]. The report Energy management for vehicle power net with fleible electric load demand [], which was presented at the 5 IEEE Conference on Control Applications brings MC p as a control strategy for a hybrid electrical vehicle. A very basic approach on how MC shold be sed as the ECA is presented. The literatre Compter controlled system handles compter controlled systems in general. Mch theory behind how to discretize systems, choose sampling time etc. Some theory on cost fnctions can also be fond in this boo [3]. The docment Conceptal power split control fnctionally ower system modeling [] is the most important docment provided by Volvo Technology. It describes mathematically the HCEV, i.e. for eample the generator, the battery etc. and also the system constraints. These eqations form a grond for the models sed in this wor. redictive control for energy management in all/more electric vehicles with mltiple energy storage system [5] is an article from the IEEE which describes a MC approach for a car with mltiple batteries. This article brings p some sefl thoghts regarding the control strategy, i.e. describing the cost fnction and how to define it Motivation to the wor Bilding real HCEV is very epensive and time demanding, and withot etensive preparations, projects can trn ot less sccessive. Therefore system modeling and simlation is mostly necessary in order to prodce good prodcts that are competitive on the maret. As a first step to introdce an HCEV base on this EMA from the Volvo grop on the maret, this report focs on developing a first simlation eperiment that incldes a model of the system and a MC reglator that controls the dynamics, and to compare this to other ECA with the same plant. The following advantages are potentially present when MC is sed in HCEV. MC as the ECA: As mentioned earlier in this paper, the literatre srvey has lead to the conclsion that there have not been any ECA based on MC reported for system similar to the one present, and this is one of the main driving forces from the Volvo grop in this matter. With the introdction of a MC controller as the ECA, a more intelligent controller is applied compare to, for eample, a I reglator. MC characteristics: With the se of MC, some new ideas are created in order to control the power split, which are then followed with theoretically and preposteros advantages. The most important ones are mltiple inpt mltiple otpt and constrains handling. Constrains handling can be achieved as well with an arbitrary ECA with careflly tning, bt to maimizing the performance, the process shold rn as close as possible towards the constraints in the system withot violating them and this can be achieved with the MC controller. Avoid high engine speeds: Becase of the predictive featre in MC, one idea is to completely remove the high engine speed modes that consme very mch fel in the ecavator. An illstration of this can be seen in Figre 3, where the prediction horizon is defined as Hp.

11 Minimize rapid engine power changes: One strong advantage with the MC featre is to minimize the rapid power change that can tae place when operating the ecavator. This is achieved by predicting the ftre power demand and ths, creating smoother power changes in the system. Real world adaptations: The reglator can be optimized for real world cycles and not only on predefined driving cycles. This is becase the ECA is based on a model described by physical and mechanical eqations of an ecavator. The ECA can also be adapted for other ecavator sizes and types. Fast online It is possible to create an eplicit MC controller which is mch faster then an online base controller de to the se of pre-calclated control laws. Dring the simlation, investigation is only needed to find the right polytope see Section 4.. for the crrent states and inpts to find the appropriate control law. This reslt in a mch faster controller compare to an online controller, where all models mst be evalated dring the prediction horizon at each sample. Figre 3. Simple illstration on how the predictive featre can be sed in a MC controller, Hp stands for prediction horizon Control goals The objective of this wor is to control the split between the primary and secondary power nit in a hybrid ecavator. The idea is to implement a model predictive control based control-techniqe with focs on the se of optimal control techniqes finite in the time horizon in combination with a nown driving cycle to derive a control algorithm. The system shold at all time satisfy the physical constraints on the inclded variables in an ecavator. This is done by operating the ecavator at a preferable engine mode and at the same time satisfies the power demand in the machine. The two power spplying nits, i.e. ICE and the shold cooperate and form an optimal power spit which minimizes the fel consmption, bt at the same time satisfy the power demand and other constraints in the system. 9

12 4. THEORY AND TOOLS In this section, the sed theory and tools are presented and smmarized. The theory is divided into three major sbsections; iece-wise affine system WA, MC and MC for WA system. After all theory has been described, the sed tools are highlighted with a following description. 4.. iece-wise affine system All systems described by hybrid dynamics are nonlinear. Assme that the derivatives in the dynamics denoted states, in are given by a nonlinear fnction, f,, of the states and some sort of inpt denoted in. Frthermore, the dynamics in the system that is not denoted states or inpts are defined as otpts y in and are given by a nonlinear fnction g,. & = f, y = g, Figre 4. Illstration on a nonlinear plant with inpts and otpts y. If the eqations above were linear or wold be made linear, the reslt wold be a linear model of the dynamics, i.e. a linear model of the plant. Many control problems are based on this plant bt with a hybrid system described mathematically, this becomes infeasible. In a hybrid system, one mathematical description is not sfficient to cover the complete behavior of the hybrid plant and mltiple mathematical descriptions are needed. For eample, in a hybrid system there eist efficiency factors on the, where the variable changes de to if the shold be charged or drained. This leads to mltiple mathematical descriptions as shown in 3-4, where the inde i stands for the different descriptions. & = f, 3 i y = g i, 4 The illstration on a nonlinear plant with three different models is shown in Figre 5. Figre 5. Illstration on a nonlinear hybrid plant with inpts and otpts y

13 If the system above wold be linearized, i.e. create mltiple linear models from the nonlinear fnctions, the reslts wold be a piece-wise affine system WA Linearization The linearization of the nonlinear fnctions is done by sing the first order Taylor epansion, where the approimation of the nonlinear fnctions is applied arond some specific points in the system. These points are denoted linearization points and form the state-space representations as will be seen frther down in this section. With the Taylor epansion applied to the epressions in 3-4 the reslt becomes linear eqations arond a linearization point, as shown in 5-6.,,,, f f f f i i i i = & 5,,,, g g g g y i i i i = 6 The linearization points, as can be seen above, are the states and the inpts in the system and these approimate the nonlinear fnctions g f, arond constant vales. With other words, if a nonlinear fnction wold be linearized arond a specific vale, the created linear fnction wold be more accrate if the evalation wold be close to the linearization point. This, of corse, is also dependent on what grade on the nonlinearity of the fnction. The offset terms in 5-6 are not preferable in this approach of WA system and therefore the epressions are rewritten to 7-8. This is done by mltiplying the parentheses and gather all constant terms in one parameter G F,, as in 9-.,, i i i i i i F B A F f f = & 7,, i i i i i i G D C G g g y = 8,,,,,, d f d f d f F i i i i = 9,,,,,, d g d g d g G i i i i = The reslt from this rewriting is the well-nown state-space form with two constants inclded. These constants are called the affine terms. The partial derivatives of the fnctions g f, are gathered together in jacobian matrices, where the elements represent all the thorogh partial derivates evalated in the linearization points. An easy eample on this is to say that a nonlinear system is described by the eqations e.-e.; = sin & e. y = e. The system eqation in this eample is clearly nonlinear and the linear matrices from the first order Taylor epansion become; F B A = = cos sin & e.3 G D C y = = e.4

14 A = cos B = F = sin C = D = G = cos o = = sin cos e olytopes One cold see a iece-wise affine system as a sort of gain schedling. Gain schedling systems, and ths WA systems are based on mltiple models of the plant and where each model describes the dynamics of the plant in a certain region. This region is bilt p with a nmber of polytopes which divides the different state-space representations. Frthermore, these polytopes are defined by the states and the inpts to the system, i.e. for a given vale of, there eists a polytope which is connected to a linear model of the plant arond a specific dynamic behavior. In the MT the creation of these polytopes are made with so called gard lines which define the interval of the region. These are called polytopes and the definition can be fond below, which is taen from the MT manal [7]. olytopic or, more general, polyhedral sets are an integral part of mlti-parametric programming. n Definition polyhedron: A conve set Q R given as an intersection of a finite nmber of closed half-spaces; Q = R n Q Q c, is called polyhedron. { } n Definition polytope: A bonded polyhedron I R I = R n c, is called polytope. { } The theory behind polytope can be fond in [7], bt as a short smmary and in this case, polytopes represent different sbspaces in the definition area for the states and inpts. When the system dynamics are in a specific polytope, then there eists a linear representation of the plant which describes the dynamics approimately. Also, polytopes are always limited in there definition area and conve. The dynamical behavior of a general class of system is captred by relations of the following form in the MT notice here the discrete time; = Ai Bi f i 3 y = C D g i i i Each dynamics i is active in a polyhedral partition bonded by the so-called gard lines; gardx gardu gardc 4 i i i For each i there eists a specific state-space representation defined by these gard lines in 4. In this wor, the dimension of the whole definition area for the polytopes is of an order 7 three states and for inpts as will be defined later on. How these polytopes are divided in space is of corse impossible to show graphically bt as for comprehension an easy eample is stdied. Assme there eists a system with only two states and the different dynamics only depend on the vale of these states, i.e. in 4 gardu i = [ ]. In Figre 6 three polytope have been plotted to illstrate where the different models are active. If in Figre 6 has a vale in 3 7 and in Figre

15 6 has a vale in, then model is active area colored red in Figre 6. Frthermore, if the states are in the set 7 and, then model is active area colored green. At last, if the vales of the states are in the interval 3 and Model 3 is active area colored ble. Figre 6. An eample of 3 polytopes which describes different dynamics in a plant. 4.. Model predictive control The method Model redictive Control MC, which shall be the applied control algorithm, has some very good aspects that sit this problem well, see Section 3.4. MC is of natre very good when handling mltiple-inpts mltiple-otpts MIMO and tae constraints on inpts, states and otpts into accont. The control strategy in the MC approach is that it ses a receding control horizon. This is becase the MC controller solves, at each sampling instant, a finite horizon optimal control problem. The calclated control signals in this way are sed differently compare to other control strategies, the first vale of the reslting optimal control variable soltion is applied to the plant and the rest are discarded. Then the same procedre is repeated at the net sampling interval with the prediction horizon shifted one step ahead, as with the strategy implies, i.e. receding control horizon. A short theoretical eplanation on MC follows, bt a more detailed version can be fond in [4]. The MC algorithm ses a so called prediction horizon and a control horizon to compte an optimal control signal. As stated before, a finite horizon optimal control problem is compted at each sample and the first control signal calclated is then applied to the plant while the rest are discarded, and then the procedre starts all over again at the net sampling interval. The idea is shown in Figre 7, where the controlled otpt ẑ shold follow some reference signal r. Notice also that H p stands for prediction horizon and H stands for control horizon. The prediction of the controlled variables ẑ is performed over an interval with length H p samples. Frthermore, the predicted control variables û are predicted dring the length of the control horizon and after that it stays fied dring the rest of the prediction 3

16 horizon. The compleity in the system increases rapidly when the control horizon is increased. This is becase the decision variables in the optimization problem increase with H. In most cases the control horizon is smaller then the prediction horizon H < H. p Figre 7. Illstration of the idea of MC. Here r is the set point trajectory, z represents the controlled otpt and the control signal. The graphical illstration is taen from []. In Figre 7 the controlled otpts are denoted z, bt in this wor the controlled otpts are denoted y. The optimization calclation in the MC algorithm incldes a cost fnction which is to be minimized and the general epression for this is stated as followed; H p H w i= H w H J = yˆ i r i Δ i 5 Q i= Here y ˆ i are the predicted controlled otpts at time and Δ i are the predicted control increments at time, and the matrices Q and R > are weighting matrices. The weighting matrices are assmed to be constant in the prediction horizon. The parameter denoted H w in 5 defines the first sample inclded in prediction horizon. This featre maes it possible to shift the control horizon when solving the optimal control problem. In this wor, this parameter is set to one H w =, i.e. the prediction starts one sample ahead from the crrent one. The cost fnction defined in 5 penitilize deviations of the controlled variables as well as variations in the control signal. A qadratic norm is sed in the fnction, which is the most common approach. This cost fnction in combination with a linear system yields a finite horizon LQ problem A simple eample on MC A simple eample on how the MC calclations proceed is shown in this sbsection. In this eample, the inpt to the system is on the form Δ, which is the increments of the control signal instead of the ordinary control signal. This is becase it is mch easier to show how the MC strategy wors theoretically. Notice that in the presented problem in this wor, the control approach is done with the ordinary control signal, bt in general, the increment form is more common. Assme a state-space system described by the simple form in e.. This system represents a doble integrator in discrete time. A physical comprehension of this ind of system is for instance problem based on vehicle steering control. R

17 =.5 Δ = A BΔ e. y = Δ = C Notice that this state-space form is an eample and the throghot elements in the matrices are not based on any dynamical behavior. In this eample we se a basic cost fnction which controls the otpts y to a reference trajectory and the increment inpts shold forced to zero. Also in the cost fnction below we say that the prediction horizon is and the control horizon is. Notice that the horizon nits are in nmber of samples. yˆ i r i Q i= i= V = Δˆ i e. Now the cost fnction cold be rewritten so that the sms are replaced by vectors and the qadratic norm; T T V = Y I Qe Y I ΔU Re ΔU e.3 Where Y is a vector containing the predicted otpt signals from the crrent sample to the second predicted sample H p = and Δ U contains the predicted control signal from the crrent sample to the first sample H =. yˆ Y =, Δ U = [ Δˆ ], Notice here that for eample y ˆ denotes the yˆ predicted otpt one sample ahead from the crrent sample. 5 Assme frther that the reference vector is constant in time; i.e. r = r =, r I = and the weighting matrices are Q = r and R = [.], i.e. penitilize the otpts eqally and the inpt less. This means that deviations from the otpts are controlled harder than the increments on the control signal. R The weighting matrices mst also be etended when the cost fnction is rewritten as followed; Q Q e = and R e = [ R]. Q Now the predicted otpts are replaced by the help of the linear eqations stated in e.. This is done by shifting e. in time to get epressions for y ˆ and y ˆ as fnction of the crrent state and the predictive control signals in time; yˆ = C = CA CBΔˆ yˆ = C = CA CBΔˆ = = CA CABΔˆ This leads to a new epression for the otpt vector in the prediction horizon; Y = Ψ Θ ΔU e.4 CA CB, where Ψ = and Θ = CA. CAB Now an epression is created that represents a tracing error, meaning the difference between the ftre target trajectory and the free response. With other words, the response that wold occr over the prediction horizon if the etended control vector Δ U e is set to zero. ε = I Ψ e.5 5

18 Assme that the crrent state is given by; = 4, then the eqation e.5 can be calclated; = = ε. If this error wold be zero = ε, then the correct optimal control signal wold be zero = Δ U e, bt this is not the case right now apparently. If the otpts were following the reference, then no change in the control signal shold occr. The cost fnction in e. cold now be rewritten with the se of e.4 and e.5 as; [ ] [ ] [ ] [ ] [ ] U R Q U Q U Q U R U U Q U U R U U Q U V e e T e T T e T e T e T T T e T e T T T T Δ Θ Θ Δ Θ Δ = = Δ Δ Δ Θ Θ Δ = Δ Δ Ψ Δ Θ Ψ Ψ Δ Θ Ψ = ε ε ε ε ε ε ε A simplification on the cost fnction derived above reslts in a qadratic eqation; U U U const V T T ΗΔ Γ Δ Δ = e.6 In this qadratic eqation e.6 the parameters are defined as followed; Q const e T ε ε = = = T = = Θ Γ = T e T Q ε = = Θ Η = Θ T T R Q This finally brings the eqation e.6 to a convenient epression e.7; V Δ Δ = = e.7 The prpose was to minimize the cost fnction in e. and therefore a minimm shold be fond for the fnction stated in e.7. This is done by finding the gradient of V in e.7 and set it to zero. The derivation is done with aspects on the control variable Δ and the reslt can be seen in e = Δ = Δ = Δ V e.8 The optimal incremented control signal that is sent to the plant in this sample is eqal to e.9;.6494 = Δ e.9 This signal mst be maniplated before entering the plant to form the ordinary control signal. The calclation in e.8 becomes very simple becase of the short prediction horizon and the short control horizon. If a larger problem shold be calclated, a least-sqare method is applied to find the optimal control signal which forces the fnction as close as possible to its minimm, bt at the same time satisfy all constraints in the system.

19 To garantee that a minimm is achieved for the cost fnction, the second derivative is investigated. If the second derivative of a fnction is greater then zero, a minimm is fond. Differentiating the gradient Δ V again with respect to Δ gives the second derivative of the fnction, as can be seen in e.. All matrices are ept as symbols to show the conditions for a minimm on the cost fnction. V T = Η = Θ QΘ R e. Δ If we assme that the weighting matrices are Q and R >, then this garantees a minimm for the cost fnction Hard constraints Defining constraints is an essential part of the MC control strategy and it is this that maes the MC approach sch a favorable ECA. In a hybrid system there eist dynamics that need to be bonded in some certain limits, which is based on physical limitations. There are two inds of constraints, hard constraints and soft constraints. In this first sbsection the hard constraints are defined and after that the soft constraints are introdced. According to the physical limitations on the dynamics, mch of the thorogh variables in the system are bonded with hard constraints. In the MT toolbo constraints can be defined on the states, inpts and otpts. The meaning of hard constraints is to bond parameters in the system in specific regions and really maes sre that those variables are constrained by physical bondaries. The cost fnction introdced in the previosly sbsection 5 shold be minimized in the prediction horizon bt at the same time satisfy all the constraints in a system. The constraints in the MC approach are generally on the control signals and the otpts, bt with the se of the MT toolbo constraints on the states are also introdced, these are defined in 6. y min min min Δ min y y ma ma ma Δ ma Hard constraints are, as the name implies, constraints that need to be flfilled at all time, i.e. the vales of the states, inpts and otpts mst be in the defined intervals when the optimal control problem is calclated Soft constraint Soft constraints are more generos constraints then hard constraints. These constraints mae the hard constraints to be violated at all time bt the violated variable is added in the cost fnction with a weighting vale to bring it bac into the right interval. An eample is shown in Figre 8, where the variable in the simlation has a soft constraint and this is violated as the figre implies. 7

20 Figre 8. Shows the parameter with soft constraints and if the bondaries are violated, this parameter is added in the cost fnction to bring it bac in the right interval. One cold see the soft constraints penalty as same as the variables in the ordinary cost fnction 5, which penalize deviations from a reference vale. This reference vale is the bondary vale on the variable and if this variable eceeds the border frther, it leads to a greater vale to minimize Model predictive control for piece-wise affine system Model predictive control for WA system has a higher compleity than a MC based on a LTI system. In a LTI system the MC is only applied to one plant model compare to a WA system, where the optimization problem is compted for all inclded linear models. At each sample, when an optimal control signal shold be calclated, the MC controller mst calclate an optimal control signal for each combination of linear models in the WA system. Notice here that this involves an online MC controller, which is sed in this wor. Motivation on this can be fond frther down in this wor, at Section 6. The best combination of the linear models prodces the optimal control which is applied to the plant. For eample, if a WA system is defined by si linear models and the prediction horizon is set to two, the MC controller mst calclate 6 = 36 different control signals before choosing the best. In this eample mentioned, the worst case scenario is presented, bt this never occrs. This is becase some of the combinations become redndant and can be discarded. Then again, the comptational time increases rapidly with aspects to the prediction horizon, control horizon and the amont of linear models in the WA system. The sampling time is also highly depended on the comptational time, which can be easily nderstood; de to at each sample, an optimal control signal shold be calclated. These parameters are very important to be considered before developing a MC controller. With the choice of a MC controller where the WA model incldes many linear models and a long prediction horizon, the reslt becomes better when controlling a nonlinear plant, logically. Bt this also leads to an increase in comptational time. A trade off mst be made between how well the WA system shold approimate the nonlinear plant, how good the MC controller shold be pred. horizon, ctl horizon and how mch CU capacity is available comptational time. The constraints in a WA system is defined as same as in a LTI system, i.e. the constraints are defined as constant vales for the different variables.

21 4.4. Matlab and Mlti-parametric toolbo All modeling and simlations are preformed in the compter program Matlab [6] and more specific Simlin, which is a simlation tool. With the MC as the ECA and the hybrid dynamics in the system, a toolbo with these preferences shold be sed and this is denoted Mlti-arametric Toolbo MT. In the following section a smmary on the Matlab commands and the sed featres in MT are highlighted Matlab and simlin The software Matlab is a mathematical and simlation program, which is sed throghot this wor. In Matlab there eist a great nmber of different toolboes and sbprograms that handle different ind of control problems. The toolbo sed for this hybrid system is the MT in combination with Simlin. In simlin the ser can simlate the behavior when the ECA is applied to the plant and evalate the reslts graphically Mlti-parametric toolbo The MTs [7] prpose is to handle system where the mathematical description is changing de to the dynamic, i.e. for eample hybrid system. Two different ind of system are compatible in the MT. These are LTI linear time-invariant system and WA system. As stated before, the WA system is the one sed in this wor. As a smmary, the sed tools to create a MC reglator for a WA can be divided into two major parts. The first part is to define the WA system as the theory implies in Section 4.. This is done throgh creating linear models of the nonlinear plant and divides them by help of gard lines. The constraints in the system are also defined here. Only constant constraints can be defined in MT and this cold lead to a limitation when handling this ind of system, more on this in Section 5. The second major part when creating the controller is to define the problem strctre. This involves defining prediction horizon, norm, weighting matrices etc. When these two parts have been defined, the MC controller can be created. This controller can be either an eplicit or an online controller. An eplicit controller precalclates controllaws offline and then connects them to certain polytopes. This alternative has a faster comptation time then the online reglator, de to the online reglator needs to calclate all matrices at every sample. The MT toolbo is compatible with the se of soft constraints and has some featres that can be defined regarding these constraints. When creating the soft constraints, it is necessary to define penalty vale on the violation. It is the same strategy here as in the ordinary cost fnction theory, that a higher vale on the penalty reslts in higher vale in the cost fnction de to the violation and forces the variable faster into the right bondaries. One cold also set an above or below limit on the violated soft constraints so that violation on the soft constraints are penalized, bt when the second limit is reach, this constraint wors as a hard constraint that can not be violated. This can be seen in Figre 9. 9

22 Figre 9. Describing the limit vale on soft constraints before it becomes hard constraints. More detailed theory can be fond in the MT se manal [7] Maple The mathematical program Maple [7] is also sed in this wor and is needed when the nonlinear fnctions shold be derivated. When the fnctions becomes complicated and inclde many variables, an easy and convenient way is to let Maple calclate the different partial derivatives in the system.

23 5. MODELING The first part of the modeling section states the general model description of the plant and driving cycle. After this, it describes the mathematical description of the plant and how to accomplish linear representations of the system de to different linearization points and different dynamics. The two last parts describe handling of distrbances in the system and the linear model validated to the nonlinear plant. 5.. Model description When woring with simlations based on physical systems, models mst be developed to wor as a sbstitte for the real world behavior. Generally, all models are approimations of the real dynamical behavior and it is of great importance to have a model which behaves as similar to its reality. There are basically two main things that mst be introdced in order to create a control strctre and these are the plant and the reglator. The plant defines the complete system and how it behaves, i.e. incldes the ICE, generator and dynamics. This cold be denoted a model, becase it is an approimation of the real world behavior. A reglator mst eist to control the plant and its dynamics. The otpts from the plant that are needed in the control strategy are then sed as a feedbac to the reglator which then calclates a control signal. For eample, assme that the otpts are the fel consmption, rotational speed and the power demand. These shold be sent to the reglator to compte a control signal based on the vales of these otpts which then shall be applied to the plant. Frthermore, in this case, a driving cycle is also introdced to create rnnable simlations and this generates the power demand in the ecavator and is a sort of inpt to the plant. One cold see the driving cycle as an artificial operator in the ecavator, i.e. perform different tas with the vehicle. An illstration can be seen in Figre. Figre. otpts y. Description of the process, where the inpts are denoted, the distrbances d and The control signal shold control the plant as stated before and this means that this signal shold tell how mch the ICE and the generator mst prodce in power to satisfy the power demand in the ecavator, and indirectly how mch power that wold be withdrawn from the. The control signal shold describe the torqe on the ICE T Eng and the torqe on the generator T Gen. The driving cycle shold describe the power demand which stand for the power from the hydralic pmps and the electrical swing. To clarify the procedre, at each sampling instant in the simlation a vale of how mch power the ecavator needs is created and sent to the plant along with a control signal based on previos sample.

24 These inpt signals are then sed to calclate all otpt signals, where some of those are then sent bac to the reglator to perform a new control signal, i.e. to control the ecavator preferably. There eist two inds of power in the system, mechanical power and electrical power. Mechanical power is the power needed for the arms and bcet illstrated in Figre. The ICE and the generator can both prodce mechanical power which is then send to the hydralic pmps. Here the mechanical power becomes hydralic power to satisfy the hydralic actators. The electrical power can be spplied both from the generator and the. The creation of electrical power is possible becase the mechanical power prodced by the ICE can be converted to electrical power by the generator. There are two different electrical power demanders in the system; the swing system and the AUX system. After the electrical power has been withdrawn, it is sent both to the SWEM, where it is converted to mechanical power in order to rotate the pper part of the ecavator and to the constant electrical power demand from the AUX system. This cold wor on the other way arond as well, i.e. the swing system cold prodce electrical power when the pper part of the ecavator is braing. When the ecavator is braing, mechanical power is prodced in the swing system which is then converted by the SWEM to electrical power. This electrical power cold both be sed to recharge the or to operate the hydralic actators with the help of the generator. The driving cycle gives a power demand for the pmps and the swing. A complete driving cycle can be seen in Figre. The power demand for the pmps is always greater then zero no regenerative energy here and is always mechanical power. The swing power sign however, can change if the rotation braes. When the swing is prodcing electrical power negative vale on the SW_power in Figre the ecavator rotational mechanism is braing, which leads to that this part becomes a power spplier. If the sign on SW_power is positive in Figre the swing system needs power to operate the rotating mechanism, which in this case comes from the electrical power on the or the generator. What can also be seen in Figre are the different wor modes and there characteristics. For eample, when the power demands from both the mps and Swing are close to zero, the idling wor mode is operating. Another characteristic wor mode is if the Swing power is negative, then a mode with mch motions on the rotation mechanism in the HCEV is operating, which is dig and dmp. When the ecavator is traveling, the rotational mechanism is fied and this can be seen in the driving cycle as only a power demand from the pmps.

25 Figre. Combined cycle with the power demand from mps and the swing. In the beginning -8 s wor mode DigDmp is present. In the end 8-9 s wor mode Lifting is present. In the middle s the wor mode Grading is present, which is characterized by a Swing power close to zero and high power demand on the pmps. The graph is taen from [6]. In Figre below, a linear model of the real plant mst be establish in order to create a MC reglator and in or case, mltiple linear models mst be determined de to different nonlinear dynamics in the mathematical hybrid system. This illstration also shows that the first control signal compted is applied to the plant. 3

26 Figre. Description of the MC reglator. The optimal control signal is applied to the plant With this nown, an optimal power split shold be applied in order to minimize fel consmption, minimize component wear and of corse not sacrifice wor performance. All of these specifications can not be optimal at the same time, so a trade off mst be introdced to get a satisfying reslt. This demands a lot of attention and it is a decision based on many variables. 5.. Mathematical description of the plant The MC algorithm demands a state-space representation of the system in order for it to be implemented and establish this description is complsory. Mathematically, the states of the system are defined as time derivative fnctions of the states themselves and the inpts. Eqation 7 describes this, where the states are denoted and the inpts are and d. The otpts of the process can be seen in 8, which is a fnction of states and the inpts. & = A B B d 7 d y = C D D d 8 d When deriving a state-space form as seen above, the identification of the states mst be done. This is accomplished throgh investigation of the epressions in Appendi.-.6 that involves time derivatives. There are a total nmber of for time derivative epressions in Appendi.-.6 and these are the rotational acceleration on the ICE, change in the ICE pressre, change in the battery level and the rate of flow of diesel. The first three mentioned describe the dynamics in the system bt the rate of flow of the diesel describes the fel consmption in the system, which only tells how mch diesel the ecavator consmes. Therefore, the rate of fel becomes an otpt signal which depends on how mch the ICE prodces. The states are defined below 9.

27 ω = pboost 9 SoE Some variables are distingished from start and those are the control signals and the distrbances, which are the inpts to the process. The signal that comes from the reglator and ths, the signal that controls the process, is introdced in, where the engine torqe T and generator torqe T GEN corresponds to how mch the internal combstion engine ICE and the generator shall prodce, respectively. The last signal in the control vector is isengon and this Boolean signal describes if the ICE shold be on/off. This signal however, is in the initial approach removed from the model to have a less complicated problem strctre. T = TGEN isengon As mentioned above, there are some distrbances that are inclded as inpts to the process. In Section 4 there is a driving cycle that generates power demand for the pmps and the swing, and these signals are introdces as nown distrbances. These are not actal distrbances; one cold see them as inpt signals that need to be satisfied. We define this distrbance representation in and clarify that SWEM = swing electrical motor. mech pmps t d = el SWEM t In order simlate and develop an ECA for the HCEV, a model needs to be created from real dynamics in a HCEV. The model mst be bilt on a mathematical theory that describes most of the dynamics in the system. Some dynamical featres are ignored to simplify the control problem. In Table, the most important parameters and variables are smmarized. arameter hysical description Unit [ ] ω Engine rotational speed [rad/s] T Engine torqe [Nm] T Trboboost torqe [Nm] boost T Generator torqe [Nm] GEN T mps actator torqe [Nm] pmps dp ressre difference in ICE [a] I Engine inertia Constant [g*m ] p Trbo pressre [a] boost K Correction factor on gas law Constant [] corr ϕ Efficiency factor on battery [] ma E Maimm energy in battery [J] SoE Level of energy in battery [] Electrical power otpt from battery [W] el GEN Electrical power otpt from generator [W] AUX Electrical power from radio, etc. Constant [W] el SWEM Electrical power from SWEM [W] 5

28 ϕ GEN Efficiency factor on generator [] loss ower losses in battery [W] ser N cell Nmber of battery cells in series Constant [] par N cell Nmber of battery cells in parallel Constant [] R Resistance in a cell [ Ω ] cell Table. Thorogh parameters and variables in the MC approach. With all states, inpts and distrbances identified, the net goal is to form the linear eqation stated in 7. This eqation is a linear fnction of the states, the inpts and the distrbances. In -4 the three time derivative fnctions of the states can be seen. T Tboost TGEN Tpmps dp > I ω& = T T T GEN pmps dp I dp K & corr ω pboost dp > getboosttimeconsta ω ω p& = 3 boost dp K & corr ω pboost dp getboosttimeconstb ω ω ϕ < ch arg e E SoE & ma = 4 discharg e ϕ E ma There is some wor to be done in order to have all the epressions above as fnctions of only the states and the inpts. The time derivative of the rotational speed is represented by the mechanical law which says that the sm of torqe divided with the engine inertia is eqal to the rotational acceleration. Engine torqe, trboboost torqe, generator torqe and hydralic pmps torqe constitte the sm of torqe if the pressre difference is greater than zero dp. Otherwise, if it is negative, the generated trboboost torqe becomes zero and this is becase the pressre difference is defined as the difference between the steady state trboboost and the crrent trboboost at a specific time 6. For eample, if the crrent trboboost is less than the steady state boost, this means that the trbo has not enogh pressre to develop its maimal torqe and there will be a torqe loss in the system. On the other hand, if the crrent boost is higher than the steady state boost, assmptions are made that the maimal torqe is generated the whole time if this condition holds, which is a good approimation. This change in the mathematical description de to different variables dp, etc. is denoted a switch. For eample, if at first, the pressre difference is greater then zero and at the net sample the variables have been changed so the pressre difference changes sign, then a new mathematical description is active. For, the torqe for the trboboost and the torqe for the pmps mst be replaced. The eqation for T boost is stated as follows; T dp ω are maps. 5 ω dp, where and boost = ω ω Using the eqation for the pressre difference, the trboboost torqe can be epressed with only the desirable variables; dp = getboost T, ω p boost, where getboost T, ω is a map. 6

29 At last, the generated torqe from the hydralic pmps is replaced with eqation 7; isengon = False Eng = off T = pmps pmps 7 isengon = Tre Eng = on ω If 5, 6 and 7 is sbstitted into, the reslt becomes a fnction of only the states and the inpts, which is presented frther down in this section at. The rewriting of the time derivative on the trbo pressre 3 are made by sbstitting dp and ω&. De to the derivitation of the rotational acceleration, eqation 39 can be derived by the combination of 38 and. The last epression, which describes the time derivative of the SoE, incldes two variables that needs to be replaced and these are the power in the battery and the efficiency in the battery ϕ. To derive an epression for 4 the law of Kirchoff is sed. The sm of all crrents in the system can be defined as; I I GEN I AUX I SWEM = 8 Where I, I GEN, I a, I SWEM are the battery crrent, generator crrent, AUX crrent and Swing electrical machinery crrent, respectively. This is illstrated in Figre 3. I I GEN I SWEM I a Figre 3. Crrents in electric node. This eqation can be changed to an epression for the sm of power from the different actators throgh the eqation on the rate of energy transfer; I = where is denoted the power and U the voltage. 9 U These combined form the sm of the powers Notice also that the voltage is the same in Figre and can be cancelled; el el = 3 GEN AUX SWEM Now the only parameter that needs to be sbstitted is the electrical generator power; mech GEN ϕgen motor operation : GEN > el mech GEN = GEN generator operation : GEN ϕgen, where ϕ = getgeneff ω, T is a map. GEN GEN 3 The epression for the mechanical generator power is sed to replace the variable in 3; mech GEN = TGEN ωgen = TGEN ω 3 This is possible de to no gear ratio between the generator and the engine, i.e. ICE monted directly on generator. To conclde, the reslt of combining eqations 3, 3 and 3 becomes; el = T ω getgeneff ω, T 33 GEN GEN AUX SWEM 7

30 The last variable sbstittion is the efficiency of the battery ϕ, which can be seen in 34. < > charge mode discharge mode, η η η ϕ 34 The η is defined by a few eqations below 35-37; loss = η 35 = loss U R, where ser N cell SoE getcell U = is a map on the battery voltage. 36 cell par cell ser cell R N N R = 37 Throgh these sbstittions the three time derivatives states can be defined as nonlinear fnctions depended on states, inpts and distrbances Notice here that the mathematical description changes de to some certain condition. > Ω Ω Ω Ω = / /,, dp I d dp I d & 38 Ω Ω > Ω Ω Ω Ω Ω Ω =,,,, dp d I K dp d I K corr corr & 39 Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω = ****,, ***,, ** /,, *,, ma ma ma ma AUX ser cell par cell cell AUX AUX ser cell par cell cell AUX AUX ser cell par cell cell AUX AUX ser cell par cell cell AUX E d N N R d E d N N R d E d N N R d E d N N R d & 4 As can be seen in 38-4, the mathematical description contains switches. The epressions changes for eample if the generator is operating as an engine generates torqe or as a generator transform mechanical power to electrical power. Additional theory can be fond in [8].

31 As above, the otpts mst be a fnction of only the states and the inpts. A straight-forward algebra combination from the epressions in Appendi. leads to the eleven otpts epressions in eqation 4-5. The otpt vector contains both all states and inpts, bt also new signals that need to be controlled or constrained. This otpt signal represents the first state in the model which is the rotational speed on the ICE. y = ω = 4 The second otpt signal is the first control signal in the system, i.e. torqe on the ICE. y = T = 4 The torqe loss in the ICE de to the trbo boost is represented by the T boost. As can be seen in the eqation below, this variable is always negative or zero torqe loss, with the assmption made before. Notice here that the trbo boost torqe is eqal to zero when the pressre difference is eqal or less then zero. Ω6 Ω3, Ω7 Ω3, dp > y3 = Tboost = 43 dp The forth otpt is the second state and represents the pressre in the ICE. y4 = pboost = 44 The fifth otpt is the torqe on the pmps and it is given by the nown distrbance from the hydralic pmps d, i.e. the mechanical power demand, and the rotational speed on the ICE. If the featre engine on/off is introdced, then this variable mst be divided into two epressions. isengon = false y5 = T = d pmps 45 isengon = tre When the states were defined there eisted a time derivative fnction which was not defined as a state. This signal becomes an otpt signal instead and represents a map on how mch the ecavator consmes in diesel fel. Notice that the signal is given in [mg/s]. If the engine is off the fel consmption becomes zero. isengon = false y = m& 6 fel = 46 Ω, isengon = tre The power otpt from the is denoted and this is the seventh otpt in the model. De to the efficiency factor on the generator transform mech. power to elec. power or vice versa this epression is divided into two different eqations. Notice also that if is negative, then power is flows into the. d AUX Ω5, if GEN y = = 7 47 d AUX if GEN > Ω5, The eighth otpt is the last state, i.e. the SoE. y 8 = SoE = 3 48 The efficiency factor on the is divided into for different eqations de to if the generator is woring as a motor or a generator, bt also becase the introdctions of the absolte sign in 9. This reslts in an epression if the vale inside the absolte sign is positive and another one if it is negative. 9

32 Rcell d Ω5, * ser par AUX N cell N cell Ω 3 Rcell d Ω5, ** ser par AUX N cell N cell Ω 3 y9 = η = 49 R cell d *** ser par AUX N cell N cell Ω 3 Ω5, Rcell d **** ser par AUX N cell N cell Ω 3 Ω5, The total power demand in the system is denoted DEM and is the sm of the power demand from the hydralic pmps, the SWEM and the constant ailiary power. y = DEM = d d AUX 5 Last otpt in the model is the generator torqe, which is the second control signal. This signal cold both be positive and negative depended on if the generator shold provide mechanical power or electrical power. y = TGEN = 5 * GEN and > ** GEN and *** GEN > and > **** > and GEN 5... Evalation of the mappings The fnctions denoted omegas, Ω n, in the eqations in the previos section represent the different maps in the system. The definitions of the maps can be fond in Appendi.7 and incldes both - dimensional fnctions as well as -dimensional fnctions. These maps are created from data achieved from tests with a real HCEV and are then interpolated to create fnctions, which are approimations of the real behavior of the dynamics. Nmerical methods were applied to create these fnctions from data sheets. These were premade when this thesis wor started. As high-lighted in the net section, when the system mst be linearized, the derivative of the fnctions mst be evalated and therefore the maps mst also be differentiable. This is done with a toolbo in Matlab called the Spline toolbo. This toolbo creates an interpolated fnction from data with some very nice featres compare to for eample interpolation fnction interpol. The most important featres for this wor is the spline commands fnval and fnder, which evalates the fnction for a given inpt vale and derivates the interpolated fnction, which are needed when the nonlinear plant shold be linearized. De to the Taylor epansion described in Section 4., all the nonlinear epressions mst be partial derivated with aspect to all inclded variables, which is done with the command fnder. Frthermore, when these fnctions have been derivated, the linearization points shall be evalated to create the linear matrices in the state-space form and the command fnval does this. This leads to a very convenient way when handling the nonlinear maps and the linearization of the plant. The spline fnctions can also be plotted with the command fnplt. The maps plotted in this way can be seen in Appendi Linearization of the system As seen in eqations 38-4 and 4-5, this state-space representation is very mch nonlinear, discontinos with several maps inclded. The net step towards obtaining a sitable MC model is to apply some approimations in order to achieve a WA representation of the process.

33 This is done with the previosly stated Taylor epansion applied to all the time derivative state fnctions and the otpt fnctions. The reslts are mltiple partial derivatives of the inclded fnctions in the statespace form which is evalated in the linearization points. All matrices in 5-53 become Jacobin matrices where each and every element comprehends to a partial derivative with respect to a particlar variable and all of these can be seen in Appendi 9.8. The elements in the created matrices are denoted a...a9 for the A matri, b b6 for the B matri etc in For eample, if the first state is set to & = f, then the first element in the A matri is defined as a = f,, d and net a = f,, d and so forth. As can be seen in the nonlinear state-space representation in 38-5, most of the eqations are fairly complicated, and with the ris of hman error, all derivatives where calclated with the assistance of the compter algebra program Maple. This finally gives the linear state-space system in & ω a a b b bd & = p& boost a a b b bd = d F SoE & a3 a 33 b 3 bd 3 ω c T d T boost c3 c3 d 3 pboost c4 T pmps c5 dd5 y = m fel = c6 d 6 d G c7 d 7 dd 7 SoE c83 η c9 c 93 d 9 dd 9 DEM dd dd TGEN d i i 5 53 Becase of the system dynamics change de to the switches stated in eqations 38-4 and 4-5, most of the elements in the matrices obviosly changes as well. As smmary, the process is approimated with several linear representation de to different condition, for eample if the pressre difference is greater or smaller then zero, i.e. dp > or dp and de to the nonlinearities in the system. These nonlinearities represent the nonlinear maps and all other nonlinear dynamic in the state and the otpt fnctions Linearization points When the tas is to linearizes a system and ths create a state-space form, the most common and natral way is to approimate the dynamics arond a stationary point. A stationary point is a point in a fnction where the vale of the fnction becomes zero or close to it. In continos time this means that the derivative is zero the fnctions is the derivatives and in discrete time the vale at the net sampling time is eqal to the previosly. 3

34 This is not a good approach to the problem presented, de to how the mathematical descriptions of the dynamics are defined. As stated before, there are several switches in the hybrid system and these comprehend each to a certain operating mode. When linearizing a nonlinear system with switches inclded, the description on where the linearization points shold be, becomes very important. This is becase if this hybrid system wold be linearized arond a stationary point, then by definition, the linear fnctions are linearized arond a point that reslts in a zero in the nonlinear fnctions. Recall from the Section 5., these switches occrs when for eample dp is close to zero. After a little investigation it is fairly straightforward to see that in the eqation below taen from 3, the fnction goes to zero if dp and ifω&. Becase ω& is one of the fnctions that shold be evalated arond a stationary point and ths is zero, reslts in the variable dp to be close to zero in order to achieve a stationary point for this fnction. dp K corr & ω pboost lim & ω, p& dp boost = lim & ω, dp = getboosttimeconsta ω ω If dp is close to zero the dynamic lies very close to a switch becase of the condition dp > or dp. The third state eqation 4 also brings ot the same problem. This is becase the switch or < is inclded in the nmerator in the S o& E and leads to a stationary point that it is close to a switch. This can be seen below where the state fnction S o& E becomes zero if the variable =, i.e. close to a switch. ϕ lim SoE & = lim = E ma In Figre 3 an illstration on this problem is shown. Figre 4. Illstration on a bad choice of linearization point. The soltion to this problem stated here is to linearize arond a non stationary point and after that maniplate the Taylor epansion with a constant in order to get the desirable form in 7-8 and not be close to any switch. The constant is the one presented in 9, i.e. f i,, and this is why it is d more convenient in general to linearizes the nonlinear fnctions arond stationary points de to the fact that f i,, d. =

35 The problem now remains to get feasible and satisfying linear models that describe the different dynamics of the system. When handling problems connected to the MC approach, the most ecessive wor is to create a complete approimate model that describes the plant in a satisfying way. De to the nonlinear preferences of the maps and the inclded epressions in the state-space eqations the plant shold be linearized arond mltiple points to create an acceptable approimation. Bt with the increase of linear models in the WA system, the comptation time becomes greater and a trade off mst be introdced between achieving a good approimation of the plant and at the same time not to increase the compleity too mch Creating WA system A good model of the plant is essential in order to achieve reasonable reslts and choosing polytopes is probably the most important part to accomplish this. As for this wor, the mission was to develop linear models that describe the plant for only the presented driving cycle in Figre. To decide the nmber of linear models in the system, and ths set the approimate level in the simlations are a little bit of a try and error method. However, some things can be established from the beginning to simplify the approach, for eample, between what bondaries the variables are defined de to physical constraints. The goal is to minimize the nmber of models and at the same time get as good linear representation of the plant as possible. To achieve this objective, investigations mst be made on all the maps and the switches to see where the nmber of switching nonlinearities occrs. hysical constraints: The inclded variables in the state-space representation have all physical constraints that describe in what interval these variables can occr. For eample, the control signal T Eng can not be negative de to the physical constraint on the ICE. Another eample is the state, trbo pressre p boost, which also can not be negative becase this is physical impossible. Nonlinear maps: All the maps in Appendi.7 are inclded in the mathematical description which defines the plant. So in order to get a feasible approimation of the plant, these nonlinear fnctions mst be stdied in order to decide how many linear fnctions to introdce. In Appendi.8 the maps have been plotted and there are a total of eight maps. Switches: There are several switches inclded in the system that describes the change from a mathematical epression to another. These are important to nderstand in order to now where the most nonlinear properties occr and ths how the system shold be linearized. Becase of the fact that the gard lines in the MT toolbo are defined as linear, these switches mst be linearized as well. The switches can be fond in Appendi.9 and the total nmber of switches is for. If all switches are introdced in the dynamics the boolean variable isengon not inclded there cold be eight different combinations of mathematical descriptions in the system. Different combinations of switches if dp >={ if Gen <{ if >{ Switch }else{ Switch } }else{ if >{ Switch 3 }else{ Switch 4 } 33

36 Figre 5. }else{ if Gen <{ if >{ Switch 5 }else{ Switch 6 } }else{ if >{ Switch 7 }else{ Switch 8 } } } The different conditions on the switches in the system. It is important to have as few linear models as possible in the MC reglator and therefore it is of high priority to investigate if all switches are introdced in the presented driving cycle. As can be seen in Figre 5, that if all the mathematical descriptions are inclded, it wold lead to at least sithteen linear models in the WA system and then all of these nonlinear descriptions mst be linearized maybe arond two points, which dobles the amont of models to thirty-two. So for starters, the option to trn on/off engine is removed. It is very hard from the beginning to now if all the mathematical descriptions shold be inclded, bt a closer loo at the condition of the switches leads to some simplifications. The eqation that defines the otpt power from the battery, 33 incldes the parameter Gen, 3 and both of these define each a switch, as can be realized in Figre 4. The switches denoted 4 and 8 are highly nliely to occr becase of the reason that if Gen is positive, it taes a great negative vale on SWEM to cancel both Gen and AUX = 5W together. The eqation from 33 is shown below with the ineqality that it shold be greater then zero. el el el el > >, if = 5, > > Gen AUX SWEM [ ] el AUX Therefore, de to the nliely occrrences of switch 4 and switch 8, these are cancelled when creating the WA system. Gen SWEM AUX Gen 5.4. Sampling time The sampling time is a crcial parameter when creating WA system as an approimation of the plant. A WA system created from the mathematical eqations given in Section 5. is given in continos time. The MC controller needs a discrete WA model and therefore, the continos WA system mst be discretized. This is done with a sampling time which approimates the continos dynamic in the system to a discrete system. With a smaller sampling time, i.e. sample the system in more points, a more correct creation of the continos dynamics is made in discrete time, bt also a more comptational depended problem. After some try and error and simlation eperience of the system, the sampling time was set to.5 seconds and this holds throghot the master thesis Handling of distrbances The inflence of distrbances is of great importance in this control problem. As stated before, the distrbance represents the power demand from the pmps and the swing, which mst be flfilled at all time dring the simlation and is introdced as an inpt signal along with the control signal to the plant.

37 This however reslts in a problem when defining the model of the plant with the MT toolbo. The Toolbo is only capable of handling a constant distrbance on state pdate eqations and not on the otpts. It is essential to inclde the inflence of the power demand pmps, SWEM both in the state pdate eqations and the otpt eqations. To develop a model withot introdcing the distrbances cold perhaps wor sometime in general, bt in this case that is an impossibility. The reason is that this 5 distrbance is so vital to the system and magnitde is so great on this signal < W, < pmps < SWEM < 5 W, that it wold tae an almost impossible robst controller to handle this behavior. So the conclsion is that it mst be inclded in the model setp and this is done by etending the state vector with the distrbances. There is an option of state feedbac in the toolbo which opens p a possibility to inclde the distrbance in the model. This is possible if the state vector is etended with the distrbance as can be seen in 54. With the help of state feedbac this model can describe the same eqations for the dynamics as with the model when the distrbances are introdces as inpts to the plant, i.e. inclded in the control signal vector. ω pboost = SoE 54 pmps SWEM When defining the distrbances as states in the system, a decision on how these shold be pdated from one sample to the net in discrete time or in the continos case how the time derivative mst be made. As a first stage in this approach the distrbance in the model is assmed to be constant dring the prediction horizon, i.e. the vales of d, d are constant when compting the optimal control problem. With the etension on the state vector, an etention of the state-space matrices follows; & A = d & Bd B F [ C Dd] D G y = 56 With this strategy and with the help of state feedbac, the power demand can be inclded in the model and a graphical illstration of the approach can be fond in Figre Figre 6. Graphical illstration on the etended state vector approach. The parameter denoted _et represents the etended state

38 5.6. Model validation In the first three sections , reslts are presented for the linearized switches and the models inclded in the dynamics. In these sections the simlations have been generated by a premade simple reglator constrcted by Jonas Hellgren at Volvo Technology with the driving cycle presented in Section 4.. With the nnown nmber of linear models in the WA system, the linearization of the nonlinear dynamics in the system was made with different approimation levels. Then these different WA systems were evalated against each other to investigate what level of nonlinearities eisted. Three models have been stated in Table 3 with different level of approimation. The first one, WA has si switches inclded and as stated before in Section 5.3., the total nmber of switches was eight, bt then two of those were cancelled. Frthermore, in this WA model, the nonlinear fnctions and maps were linearized arond two points. The approimate level is the lowest for this model. The model denoted WA has also all switches inclded bt only linearized arond one point. This model shold be more approimated then WA, de to the linear representation of the nonlinear fnctions and maps are more poorly approimated. Last model to evalate is the WA 3 model and this model has only three switches inclded. The eplanation on why there only eist three switches here is becase the trbo dynamics in the ICE are completely ignored. With this assmption the switch denoted dp is removed and reslts in only two epressions that defines the switches in this WA system. WA model Nmber of models Nmber of states WA : 6 switches and linear points 7 WA : 6 switches and linear point 6 7 WA 3: 3 switches and linear point no trbo 3 6 Table 3. Defining the different WA systems. In the first eperiment, i.e. the linearization of the switches, the simlin file sed can be seen in Appendi.. To the very left in this figre the driving cycle is generated which prodce the power demand from the pmps and from the swing, bt also the EGM signal. In the center of this simlin file, the nonlinear plant is located and this bo generates all the necessary otpts from given vales of the inpts, i.e. the power demand and the control signals from the reglator. The otpts mentioned are the vales of the nonlinear switch fnctions. The premade reglator in these linearization eperiments is of a very simple natre. With the otpts now from the plant, the torqe on the generator is calclated as followed; el el SoEref SoE AUX.5 SWEM > SWEM TGen = 57 ma, ωeng The control strategy for this reglator and ths in this epression is that the first part of 57 is controlling deviations from a reference level in the battery. This reference level is set to SoEref =. 5 and says that the HCEV shold have a crrent power capacity arond half of a flly charged battery. If there eist a large deviation from this reference trajectory, the control signal on the generator T Gen becomes larger and this means that more mechanical power is transformed into electrical power to charge the battery. The second term in 57 is very straight forward and it represents that the generator shold satisfy the constant power to the electrical system in the HCEV. Finally, the last term handles the power from the swing system and it says that if the power from the SWEM is greater then zero i.e. a power demand from the swing system, then half of that power shold be satisfied by the generator. The denominator in 57 is the rotational speed on the generator which is the same as for the ICE to create the torqe signal. After the torqe on the generator has been calclated, the torqe on the ICE is calclated in a fairly easy way as can be seen in 58. It taes the difference between the generated torqe on the pmps and the torqe losses de to the trbo boost and the calclated torqe on the generator. ref T = T T T 58 Eng pmps boost Gen

39 Now the complete control signal is calclated, bt de to the fact that the ICE can not achieve a desirable torqe simltaneosly, a torqe reference signal on the ICE is created as in 58. This reference signal combined with a reference signal on the engine rotational speed is inclded to control the engine modes. The linearization of the switches In Figre 4 the definition of all the switches in the system can be seen. There are three switches from the dynamics that form different combinations of mathematical descriptions and those are the pressre difference dp, the power otpt from the generator Gen and the power otpt from the battery. As stated in a previos section 5.3. these switches mst also be linearized in order for the MT toolbo to divide the different linear models ressre difference The eqation for the pressre difference can be fond in Section 5. in eqation 6. The nonlinear preference here is the nonlinear map which defines the steady-state boost pressre Ω 3 ω Eng, TEng and this map can be seen in Appendi This map ths also the pressre difference epression is linearized arond two points and can be seen below. The nonlinear pressre difference is calclated with the predefined map which describes the steadystate pressre boost. For a given vale on the rotational speed ω Eng and the torqe on the engine T Eng, a vale on the steady-state pressre boost can be achieved and then this vale is sbtracted with the vale of the crrent pressre in the engine p boost dring the simlation. At the same time a linear representation of the pressre difference is calclated with the form shown in 59. dp linear = dp dp mdp 59 In the table below the linear eqation parameters are shown and the nonlinear pressre difference fnction is divided into two linear eqations. These two are sed when WA model is evalated and this is becase all fnctions are linearized arond two points. For WA and WA 3, only linear part in Table 4 is sed. ressre difference model dp dp m dp Linear part Linear part Table 4. Describes the linearization points and the linear eqation parameter vales for the pressre difference fnction In Figre 7 the reslts from a simlation with the defined simple reglator is shown and with both linear parts from Table 4 operating. 37

40 Figre 7. The pressre difference for the nonlinear representation ble and the linear representation red, two linear points. The linear crve red in Figre 7 follows the nonlinear crve ble in Figre 7 very well bt in the end of this simlation there occrs an offset error. This is probably de to a vale on ω Eng and T Eng that diverge a lot from the linearization points. Besides this, the reslt is satisfying with the se of two linearization points for this nonlinear fnction ower otpt from generator The nonlinear eqation for the power otpt on the generator Gen is defined in 3. This parameter defines the second switch in the system and more physically, describes if the generator shold wor as a motor transform electrical power to mechanical power or wor as a generator transform mechanical power to electrical power. If the power otpt on the generator is greater then zero, the generator is operating as a motor and if the power otpt is eqal or smaller then zero, the generator transforms mechanical power to electrical power, i.e. wor as a generator. The linearization on this nonlinear fnction Gen can be seen in Table 5 and is divided into two linear eqation for WA and the Linear part for WA and WA 3. Gen model gen gen mgen Linear part Linear part Table 5. Describes the linearization points and the linear eqation parameter vales for the power otpt from the generator fnction. The power otpt from the generator for the nonlinear and the linear representation can be seen in Figre 8.

41 Figre 8. The power otpt from the generator for the nonlinear representation ble and the linear representation red, two linear points. This graph also shows Figre 8 an offset error from the nonlinear fnction. The difference between this approimation error and the one in the previosly graph Figre 7, is that the offset error occr dring a great part of the simlation. This mean that the dynamical behavior of the generator is harder to approimate with two points compared to the pressre difference switch ower otpt from The definition on the last switch is the power otpt from the and this fnction can be fond in Section and here below; el = T ω getgeneff ω, T 6 GEN GEN AUX De to the fact that the power on the AUX system is constant and that the power from the SWEM is nown, the only nonlinearity in eqation 6 is the power otpt from the generator. This switch is defined as if the is greater or smaller then zero, and physically if the battery shold withdraw electrical power from the battery or charge the battery. The linearization shold only be applied to the first term of the eqation which is almost as same as the mech linearization on the second switch power otpt on the generator Gen. The difference is the efficiency factor on the generator and this fnction, as can be seen in Appendi 9.8.6, is very nonlinear. The linear eqations on the nonlinear fnction can be seen in Table 6. Both the nonlinear and the linear representation are plotted in Figre 8. model SWEM gen gen m gen Linear part * 4 Linear part * 4 A. Describes the linearization points and the linear eqation parameter vales for the power otpt from the fnction. 39

42 Figre 9. The power otpt from the for the nonlinear representation ble and the linear representation red, two linear points. The linear crve follows the rapid changes of the constant, an offset error occr. in Figre 9, bt when the dynamic is near In almost all switches that are approimated, an offset error can be seen when the dynamic is constant in time. This behavior depends probably on when the switch dynamic is constant in time, the linearization points are far from the crrent operating vales. Linearization of the plant As stated before, different linear models with different approimate levels are created and evalated against each other. When the linearization points were defined, a great part of the strategy was throgh investigation of the nonlinear fnctions and maps. This lead to approimate vales on the linear points, bt after that, some try and error methods was applied to constrct the WA systems. The evalation of the different WA system was made with the simple I-reglator defined in the beginning of Section 5.6. De to this, the reslt from these simlations has a good probability to be satisfying when the MC controller is introdced as well, bt not certain. This is becase the MC controller has a different control strategy then the I-reglator, which means that the system dynamics are in other regions. For eample, the simple strctre in the sed I-reglator leads to that the generator can not wor as an engine in the simlations gen > and therefore a lot of dynamical behavior is not tested. In these three systems evalated, the linearization points for the rotational speed on the engine and the battery level have been set to =, 3 =. 5. Only the wor mode dig and dmp is simlated in this wor - s and this wor mode correspond to an engine mode EGM which has constraints defining the rotational speed arond rad/s. The battery level shold also be in some strict limits and vary arond.5. Some different sampling intervals were tested in these simlations and it was shown that a sampling time eqal to.5s was too large and made the reglator nable to control the system. Bt with a sampling time of.s the WA systems cold be controlled liewise as the nonlinear plant.

43 WA The system denoted WA has twelve linear models, i.e. si switches and two different linearization points for the torqe on the ICE. This is the most comple WA system evalated and shold show best reslt regarding approimation of the plant. The linearization points for the twelve models can be seen in Table 6. Notice here that each and every one of the first si linear models describes the switches switch -3 and switch 5-7 in Figre 5. The last si also describes the switches bt in a different linearization point. WA Model Model nr 3 Mod Mod.5 - Mod 3.5 Mod Mod Mod Mod Mod Mod Mod Mod Mod Table 6. The complete WA system. In Figre the reslt is shown when the nonlinear and the linear model for the rotational speed on the ICE are controlled by the simple I-reglator. Figre and shows the same for the boost pressre in the ICE and the battery level, respectively. d d Figre. The ω dynamic for the nonlinear WA representation ble and the linear WA representation red. 4

44 Figre. The pboost dynamic for the nonlinear WA representation ble and the linear WA representation red.

45 Figre. The SoE dynamic for the nonlinear WA representation ble and the linear WA representation red. The conclsions from these three graphs are fairly satisfying. In Figre it can be seen that the linear representation on the rotational speed follows the nonlinear fnction very well. The reference trajectory for this signal is eqal to rad/s the first seconds of the simlation specific EGM and after that, the EGM switches and the reference trajectory gets a new vale. It is a little strange that after a new reference trajectory has been introdced in the system, which lies far away from the linearization point rad/s, the linear crve follows the nonlinear crve almost with the same precision as in the first - seconds in the simlation. One eplanation is probably that one of the control strategies in the sed reglator controls the rotational speed to its reference and therefore maes the WA system close to the plant. The boost pressre in the ICE is shown in Figre with satisfying reslts for the first seconds of the simlations. After this time, a new wor mode and also an EGM is introdced, which leads to a poorer approimation of the plant. The eplanation on why the pressre crve is poorer than the rotational speed crve in Figre after seconds is that the control strategy here ignores the vale of the pressre compare to the rotational speed. The last graph Figre shows the battery level and here the reslts are not so satisfying. The linear crve lies above the nonlinear crve dring the simlation with a 3-4 % average wrong. Mch attention has been made here to tne the linearization points to achieve a better reslt on the SoE, bt withot any sccess. The reason why it deviates a lot between the linear and the nonlinear fnctions for the battery level is the comple strctre of the mathematical description. The dynamical behavior of SoE incldes mltiple switches and many nonlinearities, sch as maps and nonlinear epressions. This leads to a harder tas when linearizing the variable with a small nmber of points WA The WA system is bilt p with si linear models, where each describing a switch in the plant switch no: -3, 5-7. Only one set of linearization points are sed and the approimation level shold be higher here than for the WA system. In Table 7 the parameters for the linearized system can be fond. Model nr 3 d d Mod Mod Mod Mod Mod Mod Table 7. The complete WA system. The following three graphs show the linear and nonlinear representation of the rotational speed, pressre difference and battery level, respectively. These are denoted Figre

46 Figre 3. The ω dynamic for the nonlinear WA representation ble and the linear WA representation red.

47 Figre 4. The pboost dynamic for the nonlinear WA representation ble and the linear WA representation red. Figre 5. The SoE dynamic for the nonlinear WA representation ble and the linear WA representation red. The reslts from the WA system are very the same as for WA, bt with some difference in the battery level, which is poorer in WA. This is some proof that the approimation of the dynamical behavior for the battery level becomes worse if less linearization points are sed. Notice that the difference between the linear and the nonlinear representation in Figre 5 is the greatest in the beginning of the simlation. This can be eplained with the vale of the generator torqe T Gen =, which has a low vale dring the first - seconds of the simlation. That is why WA shows a better reslt than WA, de to different linearization sets in WA one set has a low vale on WA 3 No trbo dynamic The last model evalated is denoted WA 3. In this model the trbo dynamics have been omitted, i.e. there are not any torqe losses in the ICE de to not completely developed steady state pressre. When the trbo dynamic is remove, which is a state in the ordinary description, some switches can also be removed which depends on the pressre. Also the new model has one less state eqation to approimate. In Figre 5, the switch defined as dp pressre difference is removed and this reslts in three different switches. This model is linearized arond one set of points, bt with all three switches inclded as can be seen in Table 8. The switches are the one denoted in Figre 5 as switch -3. Model nr 3 Mod Mod Mod Table 8. The complete WA 3 system no trbo dynamic d d 45

48 Liewise as previosly, the states are plotted with the linear and the nonlinear representation in Figre 6-7. Notice here that there only eist two states and therefore also two plots. Figre 6. The ω dynamic for the nonlinear WA 3 representation ble and the linear WA 3 representation red. No trbo dynamic.

49 Figre 7. The SoE dynamic for the nonlinear WA 3 representation ble and the linear WA 3 representation red. No trbo dynamic. There are some notifying reslts with WA 3 compare to the WA system. It appears that the trbo dynamic has very little impact on the ICE for these graphs. If Figre 6 is stdied, the appearance is very similar to the rotational speed in the WA system Figre 3. According to this, the approimation of removing the trbo dynamics in the plant system does not reslt in any major changes dring this simlation. The simlation reslts with this system shold not be trsted entirely, de to the limited set of dynamical behavior that follows from the sed reglator. When all switches are introdced and the vales of the variables in the system can be set to anywhere in the definition area, then the reslt of the approimations wold become poorer. 47

50 6. MC SYNTHESIS In Section 6, the control objectives are defined as well as the creation of the MC controller, which means for eample defining the WA system for the MC controller, defining the cost fnction and the constraints in the system. 6.. Control objectives In all hybrid systems there eist many efficiency factors on most of the dynamics, i.e. the generator has one efficiency factor when operating as a motor and another when operating as a generator. Liewise is it for the SWEM and for the. The best EMA for a hybrid system is when the efficiency factors are sed optimally. A diesel engine, which is the one in this ecavator s ICE, has a very low efficiency factor and there eist a lot of losses when the engine prodces mechanical power. On the other hand, a battery has a mch higher factor compared to the diesel engine. A basic strategy for the EMA is to se the battery as a power spplier instead of the ICE when the ecavator is operating. Bt this can not be done continosly becase the wold then be drained which in trn affects the drability of the battery in a negative way. The level of power in the is denoted SoE state of energy and this level shold be ept in some specific interval. Some other things to eep in mind when describing the EMA is to satisfy the power demand from the pmps and the swing with the appropriate power from the ICE and the at all time. In other words, the power given by the driving cycle mst be flfilled by the reglator and this is not straight forward becase there eist regenetive power in the system, bt also becase the ICE can recharge the with the help of the generator. Smmary of the control strategy Satisfy power demand: This is probably the most important part in the control strategy. If this is not satisfied at all time this wold lead to an ecavator not operating properly and it will not be able to perform its assignments. Minimize fel consmption: One frther step when the power demand is flfilled is to mae the ecavator consme as little fel as possible with an optimal combination of the ICE and the. This can be achieved with varios strategies and in this wor two different strategies are considered. One is to minimize the losses in the system to get as low fel consmption as possible. The other strategy is to minimize the fel consmption directly, i.e. to minimize a parameter defining the fel consmption at each sampling interval. These two are detailed presented in section Specific interval for the rotational speed on engine: For each wor mode, the rotational speed shold be bonded in a specific interval becase it is a physical property corresponding to an EGM in the ecavator. For eample, if the ser chooses to rn the HCEV in a wor mode that comprehends dig and dmp, the HCEVs rotational speed is always in a specific interval. 4 Level on SoE: A battery s lasting effect depends on how it is sed. If a battery is drained or flly charged continosly, the lasting effect is lowered and this shold be avoided. The power capacity in the battery is denoted SoE and is a nmber between -. Zero means that the battery is drained and one means flly charged. The SoE shold be in a preferable interval dring the complete simlation. 5 Constraint handling: The possibility to define limitations on different variables de to physical constraints in the system. For eample, the control signal torqe on engine and generator, shold not eceed a vale comprehended to the maimal torqe in reality for the engine and generator. The first stated control objective, denoted satisfy power demand, is indirectly controlled by the third control objective. This is becase the rotational acceleration is depended on the difference between the prodced and demanded mechanical power in the ecavator.

51 6.. Cost fnction The MC control algorithm is based on the cost fnction defined in Section 4., bt can be seen with some changes as well below see the same section for theory enlightment. From here on, the norm of the cost fnction is set to be qadratic and there eists penalty on both the states as well as on the otpts and inpts. The reason why a qadratic norm is sed is becase a qadratic representation of the cost fnction generally yields controllers of lower compleity than their / -norm conterparts. J = H i= H p = yˆ i Q yˆ i ˆ i Q ˆ i ˆ i R ˆ i y This eqation is a very essential part of the ECA, becase it tells how the dynamics shold be controlled. There eists a great amont of different control strategies based on how the control problem is formlated, i.e. if the control strctre is based on a conventional hybrid vehicle or if in this case, a HCEV. As described before, a HCEV can withdraw power from two power spplying nits, i.e. the ICE and the, and this enables a featre to redce the fel consmption with an energy control strategy, bt also monitoring other variables of interest. The first approach is to develop an MC reglator to control a WA model of the real plant and after that replaces this model with the real plant. To start with in this strategy, two linear models need to be created, one inside the MC bloc and one that acts as the plant. These two models do not have to be the same, in fact it is mch more convenient to let the MC model i.e. the model sed by the MC reglator be less complicated then the plant model to minimize comptational time. The states and the inpts to both models mst be the same, bt changes can be made on the otpts. Otpts in the plant model in or case eist becase these signals shold be inside physical bondaries constraints and to see how the process behaves. To minimize the compleity in the MC model, a lot of these otpts are removed in the beginning. Only those otpt signals that are inclded in the control strategy are defined in the MC model. When sccessive simlations with this model have been done, the otpts inclded in the plant model are introdced in order to create a reglator that taes all the constraints in the system into accont. As can be seen in the definition of the cost fnction previosly; states, inpts and otpts can be inclded. The variable denoted ŷ in the cost fnction is the otpts that are to be controlled, which incldes both the states and the otpts, becase all states are inclded in the otpt vector. Theoretically states, inpts and otpts can be introdced in the cost fnction as stated earlier, bt in the MT toolbo, the state vector is removed in the cost fnction when introdcing penalizes on the otpts. That is why the state vector shold be inclded in the otpt vector in order to cope with the control objectives stated in Section 6.. The plant has otpts that only got a visalization prpose, i.e. otpts that are created to see how the dynamics behave and with nice physical constraints. For eample, an otpt signal from the plant model and the real plant is the total power demand from the HCEV DEM. It is of no interest to let this signal be a controllable signal, i.e. let the signal be a part of the cost fnction, de to the fact that this signal alone can not create a fnctional control strategy. With other words, this signal alone is insfficient to define one piece of the control strategy and it does not mae any sense if this signal shold be controlled to a specific reference trajectory becase it is generated by the drive cycle. This signal incldes only variables that can not be changed by the reglator, i.e. the total power demand is given as a nown distrbance, and the constraints corresponding to this signal is therefore always flfilled, which is another reason to not inclde it in the MC model. Which variables that shold be inclded in the cost fnction defines the control strategy and there eists not any straightforward approach to these inds of problems. There eist infinite many ways when creating the controllable vector ŷ in the cost fnction and this is becase new signals can be applied to the MC model and old signals from the plant model can be removed. One thing to bear in mind when creating the MC model is to minimize the nmber of otpts, de to the comptational time. 49 6

52 The control objectives in the MC controller are to satisfy the power demand from the pmps, SWEM and the constant ailiary in the system. In Figre 8 the flow chart of the power is shown, where Eng and pmps represents the mechanical power and the variables, Gen, SWEM, AUX represents the electrical power in the system. Frthermore, the eqations for, Gen and Eng all inclde control signals and the signals denoted,, represents the distrbances. SWEM AUX pmps Figre 8. A simple chart of the how the different power sppliers and demanders are oriented. Note that some of the power arrows can go in the opposite direction, i.e. _SWEM and _GEN. De to the fact that this power flow is already inclded in the dynamics stated in Section 5., no new controllable otpts need to be created to satisfy the power demand in the ecavator. This is becase, for eample, the state-pdate eqation for the rotational speed is given by the sm mechanical torqes. This variable shold be in some certain interval corresponding to a specific engine mode in the ecavator more on engine mode in Section 6.3. When a new vale is calclated for this state pdate, this vale mst be in the defined constraint and therefore satisfy the mechanical power demand within certain limits. The same is it for the electrical power otpt from the, which shold satisfy the electrical power demand. As also can be seen in the flow chart in Figre 8 is how the electrical power is distribted. The mathematical term of this has already been defined in Section 5. which is defined throgh Kirchoffs crrent law. This epression tells how mch the battery shold prodce in power to satisfy the AUX system, the swing system and if the pmps mst have additional power from the. De to this, this epression shold only be considered as a constraint where the otpt or inpt power to the battery shold be in some intervals defined by a physical constraint. Other control objectives in the controller are to minimize the fel consmption in some way and to eep specific variables in desirable intervals. For eample, to control the rotational speed in the ICE to mae the ICE wor arond a preferable operating point. The cost fnction in 6 penalizes the inpt signals from a reference vale of zero. Generally and stated in the eample of Section 4.., cost fnctions ses an incremented control signal instead of the ordinary control signal. Bt in this wor the ordinary control signal is sed and this creates problems in the cost fnction. These incremented control signals mst be introdced and follows in the net sbsection Introdction of new control signal As presented in this wor, the control signals in this system are the torqe on the ICE and the generator and are denoted,. A very common way when introdcing the control signal in the cost fnction is to replace them with the difference between the crrent control signal sample and the previos sampled signal. With other words, instead of penalizes the control signal, the penalty is set to the increments or decrements of the control signal between one sample and the net. The definition on this signal is Δ and the epression can be seen in 6. Notice here that the signal is given in discrete time. Δ = 6

53 Accordantly to how the presented control problem is defined, this approach has several advantages. One of these is to penalitized large increments or decrements on the control signal, which was one of the stated goals in the Section 3.3, i.e. to minimize rapid power changes in the system. Another advantage follows from the se of zero reference signals in the cost fnction. With zero reference trajectories in the cost fnction, all variables are forced as close to zero as possible. This is not preferable on the control signal, i.e. to eep this signal as small as possible. Instead the control signal shold have the freedom to generate a vale in the interval between the physical bondaries defined by the ICE and the generator. Instead of changing the inpt signal to the one in 6, the state vector is etended frther to create otpt signals that represent these incremented vales. Then these vales are penalized more in the cost fnction and the control signal is penalized less. The additional states that need to be introdced are the control signals on the previos sample in order to form these preferable otpt signals and the new state vector is presented in 63; ω Eng pboost SoE mech = pmps 63 el SWEM TEng TGen In Section 5.5, when the state vector was etended de to the distrbances, the state space representation had to be changed in order to create the appropriate signals. It is liewise here and the tot tot new etended matrices can be fond in 64 and 65. In the etended matrices denoted A, B, F in 64, one cold see that the last state in the state vector at the net sample is eqal to the control signal at the crrent sample. Then the new otpts can be formed from 6 with the help of the new etended state vector. tot et et et A d = Bd I B F = I A tot et B tot et F tot et 64 y C = Δ Dd D I I G = C tot et D tot et G tot et 65 With this approach the weighting matri on the control signal can be set to a very small vale, i.e. a small penalty which leads to a greater freedom on this signal Approach : Minimizes the fel consmption The first approach ses a signal corresponding to the fel consmption for the ICE, denoted m fel. This signal is introdced as an otpt signal to be minimized in the cost fnction. As stated previosly, the MC model shold inclde as few signals as possible and therefore, in the beginning, the otpt vector has only three signals. These are besides the fel consmption on the ICE, also the incremented control signals introdced in Section 6... The otpt vector for this control approach can be fond in 66. 5

54 y MC y y y = y y y = m = ω = pboost = = SoE = = Δ = Δ fel Eng = = = Ω = 3, 66 The thoght is to only minimize the fel consmption over the prediction horizon with of corse handling all constraints and the other control objectives stated in Section 6.. When minimizing the fel consmption directly as this approach does, the prediction horizon shold be fairly long in order tae advantage of this control strategy. De to the rapidly increase of comptational time when the prediction horizon is set to be longer, this approach can be a poor ECA compare to other possibilities. When the MC model has been defined and the controllable signals have been inclded in the otpt vector, the MC controller shold be defined. This means that the prediction and control horizon as well as the weighting matrices shold be defined. The weighting matrices are denoted R and Q y for the inpt vector and the otpt vector, respectively. De to the se of a qadratic norm, the weighting matrices mst be semi-definite in order to garantee a minimm in the cost fnction. This means that the diagonal elements in mst be eqal or greater then zero, i.e. R and Q, in order to garantee a minimm. i > i Weighting matri for the inpt signals: R R = 67 R Q Q3 Weighting matri for the otpt signals: Q y = 68 Q4 Q 5 Q6 Q The prediction and control horizon is a parameter that is tned to a vale for this approach frther in this wor, in Section 6.4. The elements in the weighting matrices are also tned in that Section Approach : Minimizes the losses in the system When taling abot the control strategy for hybrid electrical vehicles in general the efficiency factors play an important part when the objective is to optimize fel saving. This second approach ses a new otpt signal in the cost fnction which represents the total amont of losses in the system. No other otpt signals or inpts are inclded in the cost fnction besides the incremented control signals. The strategy is to create a signal that represents the difference between the total amont of power ot from the system and the total amont of power into the system. This is the definition of the total power losses in this system. Mathematically this is shown in eqation 69 where fel defines the transformed inner power from the consmed diesel fel, stands for the withdrawn power from the battery with the efficiency factor inclded and DEM is the total power demand. inner losses fel DEM = y 3 = 69

55 In 69 there is only one variable that has been defined earlier and it is the total power demand, which is inclded in the plant model. inner The is defined in 7 where both and η are stated in 33 and 35 respectively. inner = η 7 The last variable to define in 69 is the transformed power from the diesel fel and here the se of a transformation factor is needed Q diesel. getfelcons ωeng, Teng Ω ωeng, Teng fel = Qdiesel = Qdiesel 7 The map inclded in 7 gives the amont of diesel per second, i.e. the fel consmption in [mg/s]. As probably now to the reader, all diesel engines have an efficiency factor which is fairly bad and these vary from vehicle to vehicle and fabrication brand to another brand. The transformation factor sed here has been taen from a Volvo report written last year which they have taen from table at the website Wiipedia [6], Volvo report [7]. 4 Q diesel = 4.3 J / g A straight-forward maniplation on the map vale milligram to gram is done to bring it to [g/s] in order for it to be compatible with this transformation factor and the reslting otpt signal in which to be controlled is as followed; y 3 Ω = ωeng, T d d AUX Ω ωeng, T y3 = const y 3 Ω = ωeng, T d d AUX Ω ωeng, T y3 = eng eng eng eng Q Q Ω Ω diesel diesel Ω5, AUX d const Ω5, Ω, 5 Q Q diesel diesel const Ω5, Ω 3 3 AUX AUX Ω Ω, 5, AUX d d d d AUX d d d d AUX 3 AUX AUX d Ω5, const AUX d Ω5, Ω 5 3 **** AUX d ** *** * 7 The WA model s otpts for the MC controller can now be defined and can be seen in

56 y MC y y y = y y y = Losses = = = 3 = Δ = Δ MC MC The new matrices created in this way are defined as followed; [ c c ] MC MC MC MC Dd = [ dd dd ], D [ d d ] C =, =. The elements in the matrices are partial derivatives of losses. The constants created with the linearization of the new otpt signal losses are gathered in G liewise as in Section 4., eq. The weighting matrices are defined as in previos sbsection 6.. and are tned in Section Constraints Handling physical constraints in a system are a very important part in the ECA and the MC approach maes this possible. As mentioned in section 6. the applied control strategy is to minimize the fel consmption, bt at the same time satisfy the power demand from the pmps and swing system. These conditions are controlled with the MC approach when defining different hard constraints on the inclded variables. In Figre 4, the constraints on the rotational speed for the ICE and the SoE level are shown and here it can be seen the effect on having hard constraints on those variables preferable to control. Also in this figre is how these constraints are defined and in the figre the reader can see the pper and lower limits on the rotational speed and the SoE level in the system. This means, for eample, that the mechanical power demand from the pmps and also from the generator if it shold transform mech. power to elec. power shold be satisfied indirectly with the limits on the rotational speed de to the dynamic described in Section 5..

57 Figre 9. Eample of constraints handling the ECA. The mentioned constraint on the rotational speed can not be set to constant bondaries dring the complete simlation. This is de to the time-varying preferences and this is becase the operator in the HCEV chooses engine mode by a switch in the cabin and therefore chooses how the rotational speed shold be bonded. These engine modes are denoted EGM. High engine speeds are desired for power demanding operation. Lower engine speeds are sed for more convenient operation and normally reslts in lower fel consmption. In Table 9 the EGM modes are defined and there rotational speed interval on the ICE. Engine mode I I F G G H Engine speed range rad/s Table 9. Defining the different EGM modes and corresponding rotational speed interval. Table taen from [4] bt with conversion to rad/s. This time-varying constraint on the rotational speed on the ICE leads to some problems in the MT toolbo de to the fact that only constant constraints can be defined there. As mention earlier, constraints can be set on states, inpts and otpts, bt these constraints are the same in the whole simlation and can not be changed. In the simlation with the presented driving cycle, some different EGM:s are introdced and this needs an alternative approach. In Figre 3 the soltion is presented for this problem and it ses mltiple MC reglators with different constant constraint on the rotational speed. From the driving cycle defined in Section 5., the EGM vale is generated and with a so called switch nit, the correct MC reglator is operating with the rotational speed within correct bondaries. 55

58 Figre 3. Shows the soltion to the time-varying constraint. The EGM bo generates an EGM mode that maes the bo in the center to choose appropriate reglator with correct constraint on the rotational speed on the ICE. For eample, with this soltion, MC in Figre 3 corresponds to EGM see table 3 and if this is the case, this reglator has hard constraints between 88-9 rad/s on the rotational speed on the ICE. The same method is applied to the other reglators to cover all EGM:s. De to the lac of time, only the MC controller representing engine mode was created and sed in the simlations hysical constraints in the system In Appendi 9.9 all physical constraints in the system are defined sch as for eample the bondaries on the torqe on the ICE or the generator has a limitation on how mch it can prodce. Below a table can be seen on the sed constraints in the system and all of the constraints defined in Appendi 9.9 are not inclde de to a fairly simple starting model. Signals Lower limit Upper limit Hard/Soft constraint Inpt signal: T Eng [Nm] 75 Hard Inpt signal: T Gen [Nm] -5 5 Hard State : ω Eng [rad/s] Hard/Soft State : p boost [a] 6 Hard State 3: SoE [].4.6 Hard/Soft State 4: [W].3* 5 Hard mech pmps el State 5: SWEM [W] -9* 4 9* 4 Hard State 6: T Eng [Nm] 75 Hard State 7: T Gen [Nm] -5 5 Hard Otpt : m fel [mg/s] Otpt : losses [W] Otpt : Inf Inf Hard Hard Δ TEng [Nm] Hard

59 Otpt 3: Δ TGen [Nm] -5 5 Hard Table. The definition on the constraints se in the MC reglator. Note that the constraints on state can be changed if it is a different EGM, bt this leads to a new reglator Tning De to the highly comptational preferences in this control problem some simple different MC reglators were created and evalated. The controllers are denoted MC, MC and MC 3. All these controllers are based on the strategy stated in Section 6.., i.e. minimizes the fel consmption directly. The sampling time is set to.5 and the prediction horizon is set to only de to the comptational brden for MC and MC, and prediction horizon for MC 3. Soft constraints are introdced on the most critical variables, which is the rotational speed and the battery level. The constraint on the rotational speed can be violated by most of rad/s and.5 for the battery level constraint. All three controllers have the same penalty on the control signals as can be seen in R = 8 74 In Table, the three controllers can be fond with different weighting vales and prediction horizons. MC red. horizon enalty on y enalty on y enalty on y 4 4 Q = = 6 Q = = 3 Q = 6 Table. Shows three different MC controllers with different control preferences. Q 4 Q = Q = Q = Q 4 Q = 4 MC : The prpose with MC is to control the otpts y, y 4 to a given reference trajectory dring the presented driving cycle. This means that some important variables are almost ignored in the control strategy to get a feasible rnning model de to the short prediction horizon. For eample, the fel consmption is almost not penalized at all in the cost fnction, i.e. it has the freedom to be a high vale. The most critical variables to be in their intervals are the rotational speed on the motor and the battery level, which are penalize hard in the weighting matrices, as can be seen in Table. MC : The control strategy in MC is to apply a greater penalty on the fel consmption in order to mae it as low as possible. This leads to a shorter simlation becase of the infeasibility on the rotational speed and the battery level after a while. The penalty on the first otpt in 76, i.e. fel consmption, is here set to a smaller vale then in MC to bring the fel consmption to a low vale. MC 3: The last controller is identical to MC bt with a prediction horizon of. De to the comptational time when the prediction horizon is increased by one, the simlation is also shorter with this controller. Theoretically, this controller shold be able to achieve better reslts than MC. All other otpts are penalized eqally for the three controllers and these weighting vales are very low. This is de to that these variables shold not be controlled to a specific reference trajectory and therefore cold be any vale within the defined constraints. 57

60 6.5. Eperienced problems In this section the main problems occrred are described which can be limitations on the MT toolbo or other flaws MT limitations There is a possibility to create an off-line MC controller in the MT toolbo, which leads to a mch lower compleity and hence lower comptational time in the simlations. This is a preferable option when simlating comple problems, where the MC model needs to inclde a big WA system with a low sampling time de to the hybrid plant. Bt nfortnately, creation of an eplicit controller in the MT toolbo with this dynamics is infeasible. This is becase some of the variables in the system are defined between great intervals, i.e. for eample pmps.3 W and 9 SWEM 9 W. When the MT creates a controller with these signals large intervals in the constraints, nwanted artificial constraints are prodced by the toolbo and maes the eplicit controller only defined in a few models of the WA system. With the eistence of this bg the eplicit controller s control laws are only pre-calclated on some of the linear models in the WA system and do not cover all. With some of the models in the WA system omitted, which describes a certain dynamic behavior in the system, leads to worthless model in a hybrid perspective. One soltion to this problem wold be to scale the signals in the MC model that have big interval preferences and then rescale them to recreate the right nits. This was tested dring the wor with no sccesses, which can not be eplained. The conclsion is to ignore the eplicit controller option and create an on-line controller. This leads to more comptational demanding simlations, which reslts in bigger approimations on the plant, longer sampling time etc. Another problem that occrred is in the MT cost fnction. De to the presented control strategy, the otpts need to be introdced in the cost fnction and this is an optional featre in the toolbo. Bt when the otpts are inclded in the cost fnction, the toolbo removes the state vector in the fnction, i.e. the states are not penalized in the cost fnction. This is becase the MT assmes that the states are inclded in the otpt vector when the ser wants to penalize the otpts and hence removes the state vector from the cost fnction. It too a while before I realized this and this is one reason why the states are inclded in the otpt vector Comptational time The main isse that is created with the MC approach for these systems needs trly an engineering handling. De to switches and nonlinearities in the system, different state-space models mst be applied and one trade off is created between minimize the nmber of models to be inclded compare to how good the approimation of the nonlinear plant shold be. With a higher amont of different linear models in the system a higher comptational time is reqired. Also as mentioned in the previos sbsection, the complsory se of the on-line MC controller leads to an easy MC model and control strctre. De to this, the WA model was sed as the MC model in all the presented reslt when the MC controller is operating. This WA system has, as denoted before, si linear models which one cold see as a very easy WA system with a pretty high approimation level. When it comes to sampling time and prediction horizon and control horizon, these parameters have a very high dependence on the comptational time. This was noted when different control strctres were simlated. For eample, when the prediction horizon was greater then 3-4 samples, the simlation lead to a crash in Matlab de to the insfficient memory.

61 7. RESULTS In this section, reslts are shown for the three MC controllers defined in Section 6.4 and a MC controller operating on the nonlinear plant. De to the lac of time, all simlations were eected with a constant distrbance dring the prediction horizon. The simlin file sed for these simlations can be seen in Appendi.. In this Simlin chart, the driving cycle is located at the bottom and the MC controller in the center. Notice here that the MC controller has a state feedbac as stated in Section 5.5 and that this feedbac incldes the states, the distrbances and the previos control signals. The previos control signals are needed to create the incremental control signals defined in Section 6... In the pper part in Appendi. the linear model of the plant can be fond. This is an approimation of the nonlinear plant and this was sed for MC -3. Last in this section, a MC controller similar to the controller MC 3 was applied to operate the nonlinear plant. This was done by replacing the linear plant with the nonlinear plant in the Simlin file in Appendi MC operating distrbance constant in Hp The MC controller s prediction ability on the nown distrbance is removed and replaced with a constant distrbance vale in the prediction horizon. With other words, at the crrent sample in the controller, the distrbance is set to the same vale dring the prediction horizon as it had at the crrent sample. This was stated in Section 5.5. Frthermore, the MC model inside the MC controller is fairly simple de to minimize the comptational time dring the simlations. This is done by removing some of the nimportant otpt signals stated in Section MC In this controller the simlation cold go to 5 sec, bt with a sacrificing performance on the fel consmption as can be seen in Figre 3, which is de to the low weight on the consmed fel variable. This is becase of the low penalty vale in the weighting matri for the fel consmption. If the penalty on this signal wold be increased, the fel consmption shold decrease and this is described in the net sbsection, i.e. MC. The soft constraint on the rotational speed as well as for the battery level is violated dring the simlation. This lead to a greater penalty on these variables and reslts in the controller forces them in inside the feasible intervals, as can be seen in Figre 3. 59

62 Figre 3. Reslts from MC. A power plot can be seen in Figre 3 for MC. This plot shows the total power demand dem, the otpt power form the ICE Eng and the otpt or inpt if the variable is negative on the. The physical behavior on the power distribtion can be seen here. For eample, when the power demand is low and the otpt power from the ICE is high, then some of the power from the ICE is transferred into the to recharge the battery. De to the efficiency factor on the ICE, the total power spply from the engine and the is almost always greater than the power demand. One eample of this can be in Figre 3 at roghly 43 sec, where the total power spply is greater than the power demand.

63 Figre 3. ower plot for MC. The efficiency map for the ICE can be seen in Figre 33. De to the control strategy, the rotational speed is almost always in the interval 88-9 rad/s which correspond to the engine mode operating EGM. The reslt becase of this is that the ICE efficiency only depends on the engine torqe which can be seen in the figre. A smmary on this chart is that the ICE operates most efficient when the torqe has a high vale. 6

64 Figre 33. Efficiency plot for MC MC The main goal with a HCEV is to minimize the fel consmption and in the previos controller, this was ignored. In MC the otpt representing the fel consmption is penalized mch more. The reslt of this can be seen in the lower-right sbplot in Figre 34. Compare to the fel consmption in MC at 5 sec, where it is roghly -3 l/h, this controller maes the fel consmption to roghly l/h at 5 sec. Bt de to the low prediction horizon, some of the other constraints can not be flfilled after 5 sec. in the simlation, as can be realized when investigating the vale for the rotational speed in Figre 34. This is becase the controller drives the SoE level to its lower limit and after that, the controller can not find a feasible control law. Figre 34. Reslt from MC. The power plot for MC is denoted Figre 35 and the difference here compare to the power plot for MC is the behavior. The otpt power from the is mch higher then in Figre 3 and this

65 means that the battery drives the power demanding nits more, which leads to a lower fel consmption. Notice also that the otpt power from the ICE is lower in this simlation than in Figre 3. Eng Figre 35. ower plot for MC. The efficiency map has almost the same appearance as in the previos controller, bt with the difference when the rotational speed goes otside its bondaries. This is not a feasible reslt and can be ignored, i.e. the rotational speed violates its constraint when the controller can not satisfy to achieve low fel consmption and at the same time have the rotational speed arond rad/s. Figre 36. Efficiency plot for MC. 63

66 7..3. MC 3 The last controller evalated is denoted MC 3. This controller is identical as MC bt with a prediction horizon of. The reslt on the most important variables is shown in Figre 37 with a very good reslt on the rotational speed and the battery level. De to a longer prediction horizon, the controller can be more intelligent and constrct control signal which satisfy the inclded variables constraints in a better way then in MC. Also the fel consmption is taen into accont in the cost fnction as can be seen in the lower left sbplot. Bt this controller pays more attention to getting the rotational speed and the battery level their references, that is why the fel consmption is higher here than in the reslts for MC. A prediction horizon of leads to a very long comptational time and that is why the simlation only goes to sec. Figre 37. Reslt from MC 3. The power plot for MC 3 can be seen in Figre 38. De to the control strategy this plot shows a great average vale on the power otpt on the ICE, which leads to a high fel consmption.

67 Figre 38. ower plot for MC 3. The efficiency map for MC 3 can be seen in Figre 39. Notice here that the rotational speed is in a very tight interval, which is a good reslt. Figre 39. Efficiency plot for MC 3. 65

68 7..4. MC operating on the nonlinear plant The last eperiment in this thesis wor was to simlate the MC controller operating on the nonlinear hybrid plant, which was the stated goal from the beginning. De to the lac of time, the only MC controller introdced in this eperiment had some similar characteristics with MC 3, bt with a lower penalty on the fel consmption. The reason why this controller did not penalized the fel consmption more, was to achieve a controller that cold simlate an entire interval for an engine mode. A high penalty was therefore sed for the rotational speed and the SoE to not bring these critical variables to its limitations, i.e. mae assre that a feasible control law can be fond. Also the prediction horizon was set to. The reslts from this eperiment can be seen in Figre 4-4. Figre 4. Reslts with the nonlinear plant

69 Figre 4. owerplot for a MC controller operating on the nonlinear plant. Figre 4. Efficiency plot for a MC controller operating on the nonlinear plant. What is notable from these reslts are that the power otpt from the generator, in the lower-right graph in Figre 4, almost always is negative. This means that the ICE helps to operate the SWEM and AUX 67

70 power demand or to recharge the. Evidence regarding this is that the SoE level is ept at the preferable level.5 dring the simlation. This also means that the fel consmption becomes higher compared to some of the other simlations MC -, where the battery level is forced down to its lower limit. In Figre 4, the power otpt from the is negative a large part dring the simlation, which means that power is broght into the battry, to eep it at the battery level.5. There eisted a big difference between the nonlinear plant and the WA plant simlation. This was the simlation time, which was a lot smaller when the MC controller operated on the nonlinear plant than the linear plant. The difference in simlation time shows that there are some ncertainties when sing the MT toolbo. For eample, an error denoted Algibraic loop occrred throghot the eperiments in Simlin when the linear plant was sed, bt this error was removed when the nonlinear plant was operating.

71 8. CONCLUSIONS As a smmary, the conclsions in this wor are fairly positive. This conclsion section is divided as this wor is divided. First, conclsions on the linearization of the nonlinear plant is made, i.e. creation of the WA system. Then the control strategy is evalated with conclsions on the MT toolbo. Last, prospects and ftre wor are smmarized and presented. Creating model of the nonlinear plant: Three different WA systems were created in this wor with different levels of approimation. The reslts shown in Section 5.6 are not that important and do not tell so mch of the model validation verss the nonlinear plant. Bt with the reslts achieved in Section 7..4, i.e. when the nonlinear plant was introdced, shows that the model validation was very satisfying with only one set of linearization points. This tells that the WA system can be created with a fairly simple strctre and still mae the MC controller compte optimal control signals to control the dynamics. Control strategy: If the wor had been divided into sections representing most advanced problems, the creation of a feasible MC controller wold be the dominating part. This is partly de to the development of the different control strategies, bt mostly de to the MT toolbo. Some misnderstandments followed by some bgs in the MT toolbo slowed the wor down and some of the pre-set simlation goals had to be removed. One eample of this is the simlation with a nown distrbance dring the prediction horizon. De to the se of an on-line MC controller, the prediction horizon was forced to be low. It was only feasible with a maimal pred. horiz. of before the simlation became nreasonable time consming. For eample, if the MC controller had a prediction horizon of, a simlation of only seconds with the presented driving cycle too over hors to complete. This is why most of my reslts are shown with a prediction horizon of. The conclsion on this matter is that the MC preferences are almost not sed and with a short horizon, the controller can not minimize the fel consmption to agreeable vales and at the same time satisfy all constraints in the system. Ftre wor: MC control strategy for a HCEV is mch ndeveloped and according to this wor, mch more attention cold be applied to create a good MC controller for this problem strctre. As mentioned in the previos sbsection, the comptational factor leads to simlations with a short prediction horizon. Indications have been shown that great improvement in the overall reslts can be made with a longer prediction horizon with the same control strategy as presented in this wor. Then again, reslts have been shown that the WA model can be of a simple strctre and if the prediction horizon is to be increased, then the fel consmption can be lowered. Ftre wor also incldes trying different control strategies to see which variables that need most attention. This cold be evalated faster if some of the limitations in the MT toolbo can be removed, sch as the bg when creating an eplicit MC controller. With the se of an eplicit controller, longer prediction horizon cold be possible and a more convenient troble solving approach is introdced. 69

72 9. REFERENCES 9.. References [] Jonas Hellgren, Conceptal power split control fnctionality - ower system modeling.doc, HEM report, 8. [] Jonas Hellgren, "Conceptal power split control fnctionality - Eqivavelent Consmption Minimization Strategy.doc, HEM report 8. [3] Jonas Hellgren and Jonas Ottosson, Conceptal power split control fnctionality - EMA techniqe ideas and overview.doc, HEM report 7 [4] J.M. Maciejowsi, redictive Control with constraints, earson Edcation Limited [5] M. J. West, C. M. Bingham, and N. Schofield, redictive control for energy management in all/more electric vehicles with mltiple energy storage nits, in roc. IEEE Int. Electric Machines and Drives Conf., vol., Jn. 3, pp. -8. [6] Jonas Hellgren, Total system optimization - Load profile generation.doc, Report 8. [7] M. Kvasnica,. Grieder, M. Baotic and F.J. Christophersen, Mlti-arametric Toolbo MT, 6, [8] Jonas Ottosson, Conceptal power split control fnctionality - Dynamic programming.doc, Report 8 [9] Komatss press release page, 8/5/3 [] Johan Åesson, MCtools. Reference Manal, Janary 6, [] Jonas Hellgren, Hybrid ecavator fel consmption analysis.doc, Hybrid ecavator concepts analysis report, 7--9 [] J.T.B.A Kessels,..J. van den Bosch, Michiel Koot, Bram de Jager, Energy Management for vehicle power net with fleible electric load demand, 5. [3] Karl J. Åström, Björn Wittenmar, Compter controlled systemst, Third edition 997 [4] Jonas Hellgren, Jonas Ottosson, Conceptal power split control fnctionality - EMA techniqe ideas and overview.doc, Report 7. [5] M.J. West C. M. Bingham N. Schofield, redictive control for energy management in all/more electric vehicles with mltiple energy storage nits, IEEE 3. [6] The MathWors, [7] Maple,

73 . AENDIX.. mps A. A. T pmps pmps = ω pmps ω = ω GEN isengon = False = off isengon = Tre = on.. Engine system A.3 A.4 A.5 m& fel = = T d dt ω = ω m GEN fel ω T, ω = getfelcons T, ω isengon = false isengon = tre A.6 T Tboost TGEN Tpmps = I ω& A.7 A.8 A.9 dp = getboost T ω p p p& = pboost dt boost & boost, boost dp getboosttimeconsta ω = dp getboosttimeconstb ω K K corr corr & ω ω & ω ω p p boost boost dp > dp A. T getboosttorqe dp, ω = ω dp ω dp boost = A. T SMOKE = getsmoelimit ω getboost, p p boost ma ω, T ω boost =... T ma ω ma A. T ω = getmatorqe ω smoe ma A.3 T ma, = min T ω, T SMOKE ω, p boost 7

74 .3. Electrical system A.4 I I GEN I a ISWEM = A.5 A.6 A.7 I = U I GEN = U el GEN I = AUX U AUX A.8 I = SWEM U el SWEM.4. Generator A.9 A. A. A. ω = ω GEN mech GEN ϕ el GEN GEN = T = mech GEN GEN mech GEN, η ω GEN ϕ GEN GEN η < generator mode GEN GEN GEN > motor mode ηgen A.3 η = getgeneff T, ω GEN.5. GEN GEN A.4 A.5 d dt ϕ ϕ SoE = SoE & = E, η ma η η > discharge mode < charge mode A.6 = η =... η A.7 U = getcell SoE N cellsser Derived from following relations: η = loss ser loss N cell ; = R ; R = R par cell U Ncell

75 .6. Swing system A.8 T = I ω& frict r T T SWEM SWGEAR SWING SWING SWING SWING A.9 T frict SWING = sign ω getswingfriction ω SW SW A.3 A.3 ϕ el SWEM SWEM = mech SWEM mech SWEM ϕ, η SWEM SWEM ηswem η SWEM SWEM < generator mode SWEM > motor mode A.3 η = getswemeff T, ω SWEM SWEM SWEM.7. Mapping definitions A.33 Ω ω = getboosttimeconsta ω A.34 Ω ω = getboosttimeconstb ω A.35 Ω T, ω = getboost T, ω 3 A.36 Ω ηgen = η GEN 4 TGEN, ω, ηgen GEN GEN < generator mode > motor mode A.37 Ω T, ω = η = getgeneff T, ω 5 GEN GEN GEN A.38 Ω ω = ω 6 A.39 Ω 7 ω = ω A.4 Ω 8 = η η > discharge mode < charge mode A.4 Ω = η 9 = N N ser cell par cell R cell U A.4 Ω SoE = U = getcell SoE A.43 Ω T, ω = getfelcons T, ω 73

76 .8. Graphical illstration of maps Graphical illstration of the steady-state trbo boost map: Ω ω, T 3 Graphical illstration of the timeboostconstanta: Ω ω

77 Graphical illstration of the timeboostconstantb: Ω ω Graphical illstration of the fnction in the boosttorqe: Ω ω 6 75

78 Graphical illstration of the fnction in the boosttorqe: Ω ω 7 Graphical illstration of generator efficiency map: Ω ω, T 5 GEN