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Control-oriented modeling of two-stroke diesel

engines with exhaust gas recirculation for marine

applications Xavier Llamas and Lars Eriksson

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Control-oriented modeling oftwo-stroke diesel engines with EGR formarine applications

Journal TitleXX(X):1–23c©The Author(s) 2018

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Xavier Llamas1 and Lars Eriksson1

AbstractLarge marine two-stroke diesel engines are widely used as propulsion systems for shipping worldwide and arefacing stricter NOx emission limits. Exhaust gas recirculation (EGR) is introduced to these engines to reduce theproduced combustion NOx to the allowed levels. Since the current number of engines built with EGR is low andengine testing is very expensive, a powerful alternative for developing EGR controllers for such engines is to usecontrol-oriented simulation models. Unfortunately, the same reasons that motivate the use of simulation models alsohinder the capacity to obtain sufficient measurement data at different operating points for developing the models. AMean Value Engine Model (MVEM) of a large two-stroke diesel with EGR that can be simulated faster than real timeis presented and validated. An analytic model for the cylinder pressure that captures the effects of changes in the fuelcontrol inputs is also developed and validated with cylinder pressure measurements. A parameterization procedurethat deals with the low number of measurement data available is proposed. After the parameterization, the modelis shown to capture the stationary operation of the real engine well. The transient prediction capability of the modelis also considered satisfactory which is important if the model is to be used for EGR controller development duringtransients. Furthermore, the experience gathered while developing the model about essential signals to be measuredis summarized, which can be very helpful for future applications of the model. Finally, models for the ship propeller andresistance are also investigated, showing good agreement with the measured ship sailing signals during maneuvers.These models give a complete vessel model and make it possible to simulate various maneuvering scenarios,giving different loading profiles that can be used to investigate the performance of EGR and other controllers duringtransients.

KeywordsMean Value Engine Model, Dynamic Simulation, Parameterization, Exhaust Gas Recirculation, Ship Propulsion

1 Introduction

Over the last year, maritime transport growth slowed down.However, shipping is still growing, and for the first time theestimated world seaborne trade volume surpassed 10 billiontons.1 The required technical development to achieve anoverall clean and efficient transportation in our society hasto involve the marine sector as well. The regulations thathave driven the automotive industry started several decadesago, while the regulations affecting marine diesels beganat the beginning of the last decade. Hence, the process ofreducing the environmental impact of the shipping industryis ongoing. The International Maritime Organization hasdeveloped the stricter Tier III emission limits2 on NOx, fornew vessels built after January 2016.

Low-speed two-stroke diesel engines usually propel thelargest vessels, e.g., tankers, bulk carriers, and containerships. These low-speed engines have high fuel efficiency,

however, they are also responsible for large amounts ofpollutant emissions, like NOx and SOx. The large reductionin NOx emissions enforced by the Tier III compared tothe previous Tier II regulations cannot be fulfilled by onlyimproving the combustion of these engines. Thus, newtechnologies are being developed in order to attain theemission reductions while still keeping a good specific fueloil consumption (SFOC), which is a good indication ofCO2 emissions. The two most common technical solutionsare Selective Catalytic Reduction (SCR) and Exhaust GasRecirculation, (EGR). SCR is an after-treatment technology

1Linkoping University, Sweden

Corresponding author:Xavier Llamas, Vehicular Systems, Department of Electrical Engineer-ing, Linkoping University, Linkoping SE-581 83, SwedenEmail: [email protected]

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2 Journal Title XX(X)

that removes the formed NOx during the combustion. Onthe other hand, EGR reduces the formation of NOx duringthe combustion. This is achieved by recirculating a fractionof the burned gas back to the engine scavenging manifold.As a result, the heat capacity of the gas that enters thecylinders is increased and thus the peak temperatures arereduced. The model presented here is dedicated to the EGRapplication only.

The Tier III limits are only enforced in certainNOx Emission Control Areas (NECAs), e.g., the NorthAmerican coastal area. Currently, the number of NECAsis low but it will be increased in the coming years. Thisimplies that the target NOx reduction has to be achievedwhen the ship is maneuvering to enter a harbor. This isespecially challenging for the EGR control system sincerapid transient accelerations required during maneuveringcan conflict with the desire to keep a high EGR flow rate.The low oxygen concentration in the combustion chamberlimits the amount of fuel that can be burned withoutresulting in incomplete combustion that produces undesiredblack smoke. The transient load increase is currently themain challenge for the EGR controller, and to improve itsperformance, transient EGR testing is required.

In the automotive industry, EGR can be widely testedwith the help of well-equipped engine test beds. For the caseof large two-stroke engines, the number of available testbeds is much lower. Mainly because of the lower numberof manufacturers as well as the higher cost of building testbeds for large two-stroke engines. Even if the main enginedesigners may have a test bed for research and development,e.g., the test engine at MAN Diesel & Turbo research center,the high running cost and the low time availability limitsthe number of engine tests that can be carried out. Anotheroption for EGR testing is to perform it on engines with EGRsystem installed on currently sailing ships. This possibilityis, of course, limited to agreements with the owners andtime availability at sea. Moreover, since EGR has just beenrecently introduced, there are not many ships equippedwith an EGR system available. Besides the low number ofEGR engines to perform tests on, the high cost of runningthese large engines is also an important factor that limitsthe testing possibilities. Hence, in order to overcome theselimitations, a dynamic simulation model of such engine is ahandy tool for EGR controller development. Unfortunately,the limited availability of real engine testing measurementshas also a negative impact on the available data qualityand quantity to be used for engine model development andvalidation. Further discussions about how to deal with thelow data availability are included in this article, based on theexperience gained during the complete model development.

The main objective of this study is to present and validatea dynamic model for a large two-stroke diesel engine withEGR. The proposed model follows the modeling principles

of Mean Value Engine Models (MVEMs), widely used forautomotive engine control research. Some advantages ofsuch models are the low computational complexity andthe capacity to capture the dominant system dynamics.A general overview of MVEMs and their applicationscan be found in Eriksson and Nielsen.3 In addition, shipresistance and propeller models are included in order tostudy the dynamic interactions between the engine and theship. The model’s main application is for EGR and fuelcontroller development, e.g., Nielsen et al. ,4,5 but it canalso be a valuable tool for sizing engine components and foroptimizing and benchmarking different engine architecturesduring the design phase.

1.1 Related ResearchInternal combustion engine modeling and control hasbeen an important research topic over the last decades,and a large amount of research has been carried outconcerning MVEM development, especially for automotiveapplications. An important work that lays the foundation forsuch models is Heywood.6

Engine models for large marine diesel engines have notbeen as widely studied as for the automotive case. Largemarine diesels can be four or two-stroke based engines,where two-stroke is the standard choice for the highestpower outputs. During the eighties several authors startedto model the dynamic behavior of large two-stroke dieselengines, the main focus was the shaft speed dynamics toimprove the speed governors.7,8 A special mention is givento the work presented in Hendricks,9 where the MVEMterm was coined. More recently developed MVEMs ofsuch engines are presented in Xiros10,11 and Theotokatos.12

A model for low load operation is presented in Guan etal.,13 where it is shown that a compressor model capableof extrapolating to low load is required and proposed.Further development was done in the two-stroke modelpresented in Guan et al.,14 where a zero-dimensional crankangle resolved cylinder pressure model is implemented.Simulations of this latter zero-dimensional model are usedin Theotokatos et al.15 to define extension functions tocertain engine parameters. This results in a computationallyfast MVEM that captures the effects of changes in thecombustion input signals. For the marine four-stroke case,some examples are Malkhede et al.16 and Baldi et al.17

Since EGR is a rather new system applied to two-stroke engines, not much research has been carried outregarding EGR engine modeling. On the other hand,EGR modeling has been much more investigated for theautomotive-size combustion engines. Many ideas can beadapted from this field to the two-stroke case. However,fundamental differences exist between four and two-strokeengines, which introduce new modeling challenges. Themost important is that the scavenging manifold pressure

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Llamas and Eriksson 3

in two-stroke diesels needs to be higher than the exhaustmanifold pressure. Hence, the EGR flow has to be forcedby means of a blower.

One of the first attempts to model two-stroke engineswith EGR is Hansen et al.,18 where the work carried outin Wahlstrom and Eriksson19 was adapted from the four-stroke to the two-stroke application. Further modeling workwas done in Alegret et al.20 and extended for low loadsimulation in Llamas and Eriksson.21 All three mentionedpublications had the 4T50ME-X test engine as modelingobject. More recent work is presented in Nielsen et. al.,22

which introduces a model reduction that captures themain system dynamics and is suitable to use for an EGRcontroller. In this latter publication, the modeling approachis tested for two different engines, the 4T50ME-X testengine and a 6S80ME-C9.2 engine installed in a containership. In Wang et al.,23,24 the commercial software GT-power is used to develop one-dimensional models of two-stroke engines with EGR. The effects of exhaust gas bypass(EGB) and cylinder bypass (CB) on EGR operation areinvestigated using the developed model.

1.2 Contributions

A complete control oriented MVEM of a marine large two-stroke diesel engine with EGR is described. New modelcomponents are investigated, in particular, an analyticcylinder pressure model that can capture the influenceof changes in the cylinder pressure control signals, e.g.,injection angle and exhaust valve opening and closing.The cylinder model is developed using cylinder pressuremeasurements from a container ship engine. Moreover, amodel for the heat transfer in the exhaust is proposed toimprove the exhaust temperature predictions compared toprevious modeling work.

A parameterization process, designed to deal with thescarcity of measurement data is also described. It is shownto obtain good model parameters that are capable ofmatching the measured operating signals of the enginefor a wide range of loads and engine running modes(ERM). Furthermore, the experience gathered through thecomplete development of this engine model is used toidentify important measurement signals that are currentlyunavailable but could be included in future measurementcampaigns. These identified signals can be valuable tothe research community when parameterizing the proposedmodel to other engines, and for further validation of theengine component models.

In addition, propeller and ship resistance models areinvestigated. Together with the proposed engine model,they constitute a complete simulation model of a containership. The model is ready to be used for simulation-based investigations of EGR controller performance during

typical load transients introduced when the vessel ismaneuvering.

1.3 OutlineThe experimental data used to develop, parameterize andvalidate the proposed model is introduced in Section2. Section 3 presents the modeling approach of thedifferent components, with special attention to the proposedanalytical cylinder pressure model. Section 4 describes themodel parameterization procedure. Section 5 contains themodel validation against the available measurements forboth stationary and dynamic conditions. A short discussionabout guidelines for future engine testing is included inSection 6. Section 7 introduces the propeller and shipresistance models used to simulate the complete containership and shows a simulation example to validate the model.

2 Experimental DataThe measurement data used to design and tune the proposedMVEM is taken from the different sensors mounted onthe ship engine. The sensors record the different engineoperating signals while the ship is sailing. Then, the datais post-processed to extract stationary points for the modeltuning and validation procedures. The post-processing isalso done to avoid having a repetition of operating points,which could bias the model accuracy towards the operatingregion with the higher number of points. The measuredengine operation spans a reasonably wide range of loadand operating modes. A total of 52 stationary points havebeen extracted for the parameterization process. However,the amount of different operating points collected is notclose to the usual engine mapping done for calibration ofautomotive MVEMs. Nevertheless, the available data issufficient to parameterize the proposed model as will beshown in Section 5. In addition, 40 stationary points arereserved for the stationary validation of the engine model.

Furthermore, in-cylinder pressure measurements areavailable for 24 different operating points. These weresynchronized with the rest of the measuring setupsystem to gather data for developing the new cylindermodel described in Section 3.7. All in-cylinder pressuremeasurements were recorded earlier than the data usedfor stationary point extraction. Therefore, the cylindermeasurements cannot be synchronized with the previouslymentioned extracted stationary points. Moreover, thecorresponding operating points to each in-cylinder pressuremeasurement were not long enough to obtain a stabilizedexhaust temperature measurement, so this data has onlybeen used for developing the cylinder pressure model.

For future applications, the stationary shop test datacollected during the engine commissioning should besufficient to parameterize the proposed model. This is



important since the engine model could be tuned before itis installed on the ship. In this case, this was not possiblesince certain signals required were not available, e.g., thecylinder exhaust valve opening and closing angles.

3 Engine ModelingThe modeled engine is a MAN Diesel & Turbo 6S80ME-C9.2 two-stroke uniflow diesel engine equipped with anEGR system for Tier III operation. It has six cylinderswith 0.8 m bore and 3.45 m stroke, and it can deliver amaximum rated power of 23 MW at 73.9 rpm. The maincharacteristics of the engine air path can be seen in Figure 1.The engine has two turbochargers in parallel that deliver thenecessary air. The EGR system takes gas from the exhaustmanifold and delivers it to the gas mixer using two EGRblowers.

The control volumes are depicted as rectangles, withthe corresponding model states written inside. The genericcontrol volume has states for pressure, temperature, andgas concentration. However, not all control volumes containall state dynamics of the generic model, as can be seenin Figure 1. The three control volumes in the EGR loopassume that the volume is isothermal. Thus, its temperatureis set by the inlet flows temperature. The control volumeof the turbine outlet assumes that the gas concentration isequal to the inlet gas concentration. Finally, the compressorinlet and outlet control volumes assume that the onlydynamic change occurs with the pressure. Moreover,the turbocharger speeds and the exhaust manifold walltemperature are also model states as will be discussed inSections 3.3 and 3.10.

In total, the MVEM has 41 states and 11 control inputs,shown in Figure 1. Four control inputs correspond to thecylinder model; αEV C , αEV O, αinj and Y . Four morecontrol inputs are required for the EGR system; uSDV ,uCOV , ωeb,1, and ωeb,2. Finally, three more general controlinputs are needed; utc, uEGB and uAux. The air ambientpressure, temperature and relative humidity can be eithertaken as constants or set equal to the measured signals.

The engine has four main Engine Running Modes(ERM), which are included in the proposed model.Number one is with both turbochargers running and noEGR operation. The second mode is with only the mainturbocharger running, using the cut-out valve utc, and noEGR operation. Number 3 is low EGR operation with onlythe main turbocharger and one of the two EGR blowersoperating. Finally, the fourth mode is high EGR flow withboth EGR blowers and only the main turbocharger running.

3.1 Gas Composition and PropertiesWhen recirculating burned gases, the amount of oxygen inthe scavenge gas is reduced. This increases the heat capacity

Cylinders

Wc,1WEGR

Wdel

Wcyl

Wt,2

Turbine

Compressor

Wout

ωeb,1

uCOV

Turbine

Compressor

ωtc,1ωtc,2

pc,2

Wc,2

Wt,1uEGB

uSDV

Y

WEGB

αinj

αEVC αEVO

ωeng

utc

utc

Win

pc,1

uAux

Gas

Mixerpmix Xmix Tmix

Scavenging

Manifoldpscav Xscav Tscav

Ambient

pamb

Tamb

RHamb

XEGRpEGR

Exhaust

Manifoldpexh Xexh TexhTew

Cooler

Cooler

Cooler

WAux

Tt,outpt,out

pc,in

Xeb,inpeb,in

Xeb,outpeb,out

ωeb,2

Figure 1. Diagram of the modeled two-stroke engine with themodel states inside the depicted control volumes. The diagramalso contains the location of the different control signals andgeneral mass flows.

of the gas and thus reduces the amount of NOx producedduring combustion. However, EGR has the undesired effectof lowering the amount of fuel that can be burned sincethis is directly proportional to the amount of oxygen in thecombustion chamber.

Keeping track of the scavenging gas composition iscrucial in a diesel engine with EGR. Thus, the engineworking fluid is modeled as a mixture of O2, CO2, H2O,SO2 and N2. The reason to include more species than O2

is to make the model capable of working with differentkind of fuels, with more or less SO2 content. Note thatonly lean operation is considered, and thus there is no needto consider CO and H2. Moreover, since the scavengingoxygen sensor is measuring molar concentrations, keepingtrack of the complete gas composition simplifies the processto change from mass to molar concentrations and vice-versa. The concentration vector in an arbitrary controlvolume is defined as

X =[mO2

,mCO2,mH2O,mSO2

]T

mtot(1)

N2 is not included in the concentration vector since itcan be computed as the remaining part. To ensure that themass conservation law is not violated, the mass flow of N2

entering and exiting the engine model can be monitoredto validate the numerical integrations during simulations.The gas thermodynamical properties are calculated usingthe NASA polynomials,25 and assuming a thermally perfectgas, i.e., the heat capacities of the gases are functions oftemperature and composition but not pressure. The inletfresh air concentration is calculated with the standard airoxygen concentration and the measured relative humidity(RH) at the inlet.



3.2 Control VolumesIn an MVEM, the different pipe volumes of the engine areseen as control volumes that store mass and energy. Thefilling and emptying of those volumes define the dynamicbehavior of the model. Thus, the bigger the volume is,the slower it will be to change its current states. Thelargest control volumes in a two-stroke engine are thescavenging and exhaust manifolds and these define the maintime constants together with the turbocharger rotationaldynamics. In addition, several others are added in order toavoid algebraic loops that are introduced when connectingseveral flow restrictions in series, e.g., the control volumesat the compressor outlets.

For the main control volumes, mixing, scavenging, andexhaust, mass and energy conservation laws are usedto derive the state differential equations. Note that thegas thermodynamic properties are a function of the gasconcentration and temperature, i.e., cv(X,T ) and R(X).Starting from the conservation of mass as it is done inEriksson and Nielsen,3 the change in stored mass in thecontrol volume can be written as

dm

dt=∑j

Win,j −∑k

Wout,k (2)

Considering the control volume as a well-stirred mixer, thedynamics of the concentration states can be derived startingwith the conservation of mass equation, (2), which yields

dX

dt=R(X)T

pV

∑j

(Xin,j −X)Win,j (3)

Applying the conservation of energy law will define thedifferential equation governing the temperature change inthe control volume. This derivation is done starting withthe same equation as in Eriksson and Nielsen.3 However, inthis case the specific heats are not considered as constantswhich makes the expressions longer. After rearranging theterms it can be written as

dT

dt=

R(X)TpV (A−B)− T

(∂cv(X,T )

∂X

)TdXdt

cv(X,T ) + T ∂cv(X,T )∂T

(4a)

A =∑j

Win,j (cv(Tin,j , Xin,j) +R(Xin,j))Tin,j

(4b)

B =∑k

Wout,kR(X)T +∑j

Win,jcv(X,T )T + Q

(4c)

where Q represents the heat transfer from the controlvolume to the surrounding. Note that the gas concentrationderivative, dXdt , has to be inserted in (4a).

Having the control volume temperature and stored massas states, the pressure in the control volume can becomputed using the ideal gas law. This results in a compactnotation but requires an extra nonlinear equation to obtainthe pressure as the model output. Another approach, whichis the one followed here, is to get the pressure as a state inthe control volume and thus directly available as a modeloutput. By differentiating the ideal gas law, the pressuredynamics can be written as

dp

dt=R(X)T

V

dm

dt+

p

R(X)

(dR(X)

dX

)TdX

dt+p

T

dT

dt(5)

with the concentration and temperature derivatives,(dXdt ,

dTdt ), directly substituted in (5). Note that the mass

derivative, dmdt , appears in (5) for compact notation. But itdoes not need to be integrated since it only represents thedifference between entering and exiting mass flows to thecontrol volume (2). Appendix A contains information abouthow to compute the thermodynamical properties of the gas.Moreover, it also contains definitions of the derivativesappearing in the previous expressions.

The previous equations are written for the genericcontrol volume case. However, some of the control volumesin the model are simplified, neglecting the temperatureor the gas concentration dynamics. This is shown inFigure 1, where the states are shown in each controlvolume. The simplifications are done by setting thecorresponding derivatives to zero in the previous derivedgeneric equations. Note that the heat transfer in the Gasmixer and Scavenging manifold are set to zero, while theheat transfer model in the exhaust manifold is described inSection 3.10.

3.3 TurbochargersThe engine has two turbocharging units working in parallelto supply the engine with sufficient air mass flow. Whenthe engine operates with EGR the secondary turbochargeris disabled with the use of the cut-out valves, utc.

The compressor model structure is defined as

Wc =fWc(pc,in, pc,out, Tc,in, ωtc) (6a)

ηc =fηc(pc,in, pc,out, Tc,in, ωtc) (6b)

Tc,out =Tc,in +Tc,inηc

(pc,outpc,in

) γc−1γc

− 1

(6c)

Pc =Wc cp,c (Tc,out − Tc,in) (6d)

where the low load extrapolation capable compressor modeldescribed in Llamas and Eriksson26,27 is inserted in (6a) and(6b). This compressor model has been proven to work wellfor low load simulation.21



The turbine model outputs are computed as

Wt =fWt(pt,in, pt,out, Tt,in, ωtc) (7a)

ηt =fηt(pt,in, pt,out, Tt,in, ωtc) (7b)

Tt,out =Tt,in − ηt Tt,in

1−

(pt,outpt,in

) γt−1γt

(7c)

Pt =Wt cp,t (Tt,in − Tt,out) (7d)

where the turbine mass flow and efficiency models fromLlamas and Eriksson21 are inserted in (7a) and (7b).

The shaft speeds of both turbochargers are modelstates, defined by the power balance between each turbineand compressor. The turbocharger speeds are the slowestdynamics of the model and thus the most dominant statesfor the engine transient behavior. Using the formulationdescribed in Eriksson and Nielsen,3 each shaft speed iscomputed as

d

dtωtc =

Ptηmech − PcJtc ωtc

(8)

where ηmech describes the friction losses of the connectingshaft and it is used as a tuning parameter.

3.4 Air Coolers Restrictions and ValvesAll valves in the model, see Figure 1, are modeled ascompressible turbulent restrictions with the valve openingangle as input.3

Wv =Aeff (uv) pv,in√

Rv,inTv,in

√2 γv,inγv,in − 1

(Π

2γv,in −Π

γv,in+1

γv,in

)(9a)

Tv,out = Tv,in (9b)

The relation between the effective area,Aeff , and the valveopening signal, uv , is highly nonlinear. The engine designeridentified this nonlinear relation and provided it to theauthors. The maximum effective area for each valve used toscale the static relation, is considered a tuning parameter;ASDV , ACOV , AEGB . The turbocharger cut-out valve,controlled with utc, could be modeled as a compressiblerestriction. However, in order to avoid having to includemore tuning parameters and control volumes, instead, it ismodeled as a switch that enables or disables flow throughthe secondary turbocharger.

All air coolers consist of a flow restriction, a coolingelement, and a water mist catcher (WMC). The restrictionis modeled as an incompressible turbulent restriction,3

Wac = Aac

√pac,in

Rac,inTac,in(pac,in − pac,out) (10a)

Tac,out = Tac,in − εac(Tac,in − Tcool) (10b)

where the cooler equivalent area, Aac, is not a geometricalarea but instead it is used as a tuning parameter. Theareas are named; Aac,1, Aac,2, Aac,egr. The coolant fluidtemperature, Tcool, assumed constant, and the coolingelement effectiveness, εac, is modeled as a quadraticfunction of the cooler mass flow as in Theotokatos12

εac = kac,1 + kac,2Wac + kac,3W2ac (11)

where kac are tuning parameters and the coolant fluidtemperature.

The WMC is assumed to remove all condensed waterin the flow. The WMC input mass fraction is convertedto molar fraction, XH2O,in, using the transformation givenin Appendix A. The water saturation pressure, pH2O,sat, iscalculated at the flow temperature.28 The ratio between thesaturation pressure and the input pressure corresponds tothe maximum volume fraction of vapor in the flow. Sincewe assume that the gas follows the ideal gas law, the volumefraction is equal to the molar fraction. The outlet molarconcentration is limited to the saturation value as

XH2O,out = min

(XH2O,in,

pH2O,sat(Tac,out)

pac,out

)(12)

If the water molar concentration is higher than thesaturation value, the condensed water is removed and theoutput mass flow and mass fraction vector are recalculatedaccordingly.

The restrictions from the ambient to the compressorsinlet control volume and the turbines outlet control volumeare also modeled as incompressible turbulent restrictions.The tuning parameters are named; Ain, Aout. They definethe inlet and outlet flows to the engine, see Win and Wout

in Figure 1. Where the outlet temperature of these tworestrictions is assumed to be equal to the inlet.

3.5 Auxiliary BlowersThe engine auxiliary blowers are responsible for providingsufficient air mass flow at low loads where the turbocharg-ers efficiency is low. As done in Guan et al.,13 the pressureincrease is modeled as a quadratic function of the blowervolume flow. This relation is then inverted21 to have thepressure difference as input and the volume flow as anoutput of the model.

WAux =pmix(kAux,1 + kAux,2

√kAux,3 − (pscav − pmix))

RmixTmix(13a)

TAux,out = Tmix +TmixηAux

(pscavpmix

) γmix−1γmix

− 1

(13b)

Unfortunately, there is little information about theperformance of the auxiliary blower. Only two operatingpoints of volume flow and pressure rise are specified at



the operating blower speed, and are used to adjust themodel parameters kAux,1, kAux,2 and kAux,3. The auxiliaryblower output temperature (13b) is modeled using theisentropic efficiency definition and assuming a constantefficiency value.

The auxiliary blowers dynamic behavior during startand stop is also unknown. The time it takes to acceleratethe blowers is modeled as a first order system, where thetime constant is adjusted to match the measured pressureresponse. This will be further discussed in Section 5. Thetime it takes to stop the blowers is also taken as a first ordersystem, with a distinct time constant since decelerating isfaster than accelerating. The dynamics can be written as

d

dtuAux,d =

uAux−uAux,dτAux,start

, for uAux − uAux,d > 0

uAux−uAux,dτAux,stop

, for uAux − uAux,d < 0

(14)where the value of uAux,d, follows dynamically the orderedstart or stop step signal uAux. The output of (14) is used toscale the flow of the blower during start and stop as

WAux,d = uAux,d WAux (15)

Note that when uAux,d has reached uAux during a start, itsvalue is 1 and thus WAux,d = WAux. Hence, the mass flowscaling is only active when the control signal uAux changes.

The one-way valve between the Gas Mixer and Scav-enging control volumes is modeled as an incompressiblerestriction (10a), without temperature change, using (9b),and with tuning parameter AAux. This restriction starts togovern the flow as soon as the pressure in the scavengingmanifold is lower than the pressure in the gas mixer. Whichis the normal case when the Auxiliary blowers are notenabled since no backflow is allowed through the blowers.

3.6 EGR BlowersThe mass flow provided by the EGR blowers is modeledusing the non-dimensional parameters Ψ and Φ, whichare assumed to follow an elliptical relation. This modelhas been used previously for EGR blowers.20–22 Theuse of the non-dimensional parameters is also motivatedsince they operate at varying speeds but there are onlyperformance measurements for a single speed line. Thus,the non-dimensional parameters give a convenient wayto extrapolate the blower performance at different blowerspeeds, as it is shown in Eriksson29 for automotivecompressors. The structure of the model is as follows

Web =peb,inD

3ebΦωeb

2Reb,inTeb,in(16a)

Teb,out = Teb,in (16b)

where Deb is the diameter of the blower impeller, andΦ is computed from the elliptical relation with Ψ and

the current operating conditions, as described in20,22. Theblower temperature at the outlet is assumed to be equalto the inlet since the temperature measurements indicate avery low temperature rise when the blowers operate.

3.7 CylindersTo improve the cylinder model compared to the previouspublications20,21 was one of the primary objectives of thiswork. The main interest is that the model resembles asmuch as possible the physical response of the real engineto changes in control inputs. The cylinder pressure controlinputs are; the opening and closing of the exhaust valve(EVO and EVC), the fuel injection angle αinj , and the fuelindex Y that defines the injected fuel mass flow Wf .

One option would be to implement a crank resolvedcylinder pressure model as in Guan et al.14 However, thiswill conflict with the objective to have a faster than realtime MVEM. In order to improve the cylinder model andstill keep it fast, an analytic cylinder pressure model isdeveloped by adapting the four-stroke model described inEriksson and Andersson30 to a two-stroke marine dieselengine. The six cylinders are assumed to operate identicallyso that one cycle computation will represent all six.

A diagram of the cylinder pressure model is shown inFigure 2, where the different key points are numberedin a similar way as in the ideal Otto cycle.6 Note thatthe start of compression point, point 1, depends on thecontrol input EVC. The point 3 corresponds to the endof combustion. The end of expansion and start of theblowdown process, point 4, is controlled by EVO. Point5 is the end of the blowdown process, which is definedby the fixed angle when the intake ports open, IPO.All thermodynamic properties, e.g., cp, cv , R and γ, arecalculated at the different parts of the cycle as a functionof the gas composition and temperature with the helpof the NASA polynomials.25 Appendix A contains moreinformation about these calculations.

3.7.1 Cylinder Scavenging Scavenging is the process ofremoving the burned gases from the cylinder and fill itwith unburned gas from the scavenging manifold. It is acomplicated process involving many physical interactionslike heat transfer and fluid dynamics, and it requiresadvanced models if high accuracy has to be achieved.The goal here is to capture the main characteristics of thescavenging process while keeping a low complexity and fastanalytical model.

CFD analysis of similar uniflow scavenged enginesshow that the scavenging efficiency is usually very high,and the process involves a displacement and mixing ofgases, see, i.e.,31–33. Based on CFD simulations, a simpleanalytic model for the scavenging efficiency is defined inAndersen.31 It is based on a modification of the classicalperfect displacement model.34 The proposed model also



Figure 2. Diagram of the measured cylinder pressure in thickdashed line together with the modeled cylinder pressure insolid line. The compression and expansion asymptotes aredepicted in dotted and dashed-dotted lines respectively. Thenumbers correspond to the different key points of the idealOtto thermodynamic cycle.

contains an extra submodel to account for the scavengingprocess that occurs during the push out phase. According toAndersen,31 it is important to consider the expelled burnedgases when the scavenging ports are closed but the pistonmoves upwards, and the exhaust valve is still open. Themodel equation with the volumetric and push out terms iswritten as

ηscav = a

[1 +

ρexhρscav

(1

DR− 1

)]+ b︸︷︷︸

dψvol

+ cVIPC − VEV C

Vref+ d︸︷︷︸

dψpush,out

(17)

where DR corresponds to the delivery ratio.31,34 Theparameters a and b are taken as in Andersen.31 Since thecylinder volumes are different than the original ones,31 theconstants c and d are scaled to have similar output valuesfor the push out term, dψpush,out, as the values reported inAndersen.31

3.7.2 Cylinder Mass Flow The mass flow going throughthe cylinders is modeled as is commonly done for MVEMof two-stroke engines.8,12,20 A compressible restrictiondefined as (9a) is used with a fixed equivalent area as atuning parameter, Acyl. The flow through the cylinders isnamed delivered flow, Wdel.

The fuel mass flow is a function of the ordered fuel index,Y , and the current engine speed

Wf = kf,1 + kf,2Y ωeng (18)

where the parameters kf,1 and kf,2 are adjusted with shoptest fuel mass flow measurements and kept constant. The

mass flow exiting the cylinders is

Wcyl = Wdel +Wf (19)

Note that to calculate mass from the mass flows, the enginespeed in rps is used.

3.7.3 Compression Pressure The compression pressureasymptote is modeled as a polytropic process starting atthe point 1, see Figure 2. Pressure and temperature arecalculated as

pco(θ) = p1

(V1

V (θ)

)γco(20)

Tco(θ) = p1

(V1

V (θ)

)γco−1

(21)

where the polytropic exponent, γco, is calculated withtemperature and composition at EVC and kept constantthrough the compression process.

γco = γ(T1, X1) (22)

During scavenging and push out, from the Intake PortOpening (IPO) to EVC the cylinder pressure is somewherebetween the scavenging and the exhaust pressures. This canbe observed on the cylinder pressure measurements, and itis also described in Lamas et al.33 However, to keep themodel simple, the cylinder pressure is assumed to be equalto the scavenging pressure, pscav between IPO and EVC.Thus, p1 = pscav .

Determining the temperature at EVC is more difficult.First, because no cylinder temperature measurements areavailable, and because the complex scavenging process hasa significant effect on the cylinder temperature during theopen phase. The starting temperature is computed as inEriksson and Andersson,30 a mixture of the fresh air atscavenging temperature and the unscavenged gases at theprevious cycle final temperature. Once the pressure andthe temperature are determined, the trapped mass in thecylinder is calculated using the ideal gas law.

3.7.4 Expansion Pressure A polytropic expansionasymptote is used to model the expansion process,30 seeFigure 2. The asymptote is defined as

pex(θ) = p3∗

(V2

V (θ)

)γex(23)

Tex(θ) = T3∗

(V2

V (θ)

)γex−1

(24)

where the polytropic exponent, γex, is calculated withtemperature and composition at the hypothetical point 3∗

and kept constant through the expansion process.

γex = γ(T3∗ , X3∗) (25)



This hypothetical cycle point 3∗, which defines theexpansion asymptote, is determined by adding thecombustion temperature increase described in Eriksson andAndersen30 to the temperature at point 2.

∆T =mfqLHV ηf (αinj)

cv(T2, X2)(mf +mtrap)(26)

T3∗ = T2 + ∆T (27)

With T3∗ and using the ideal gas law, the pressure canbe determined at the cycle point 3∗, see Figure 2. Theburned gas mass fraction at point 3∗, X3∗ , is computedassuming complete burning of the injected fuel and theoriginal composition of the charge trapped in the cylinder,as described in Nielsen et al.22 The efficiency term in(26), ηf , is used to account for pressure and temperaturevariations in the expansion part of the cycle depending onthe injection angle, αinj . It will be further discussed in thefollowing Section.

3.7.5 Combustion Interpolation The combustion part ofthe cycle is modeled as an interpolation between the twoasymptotes, see Figure 2 for a graphical example. Theinterpolation is done with the help of a Vibe function.30

PR(θ) = 1− e−a( θ−θo∆θ )m+1

(28)pcyl(θ) = (1− PR(θ))pco(θ) + PR(θ)pex(θ) (29)

The Vibe function has four parameters that need to beestimated; a, m, θo and ∆θ. Note that the Vibe functionis over-parameterized,3 since different combinations ofparameters can give the same output shape. To avoid this,the parameter a value is fixed to 5.8. Separated leastsquares problems are solved for each measured cylinderpressure to obtain the remaining three Vibe parametersand the efficiency term, ηf , for each operating point. Thenthe relations between the obtained Vibe parameters andother model signals and inputs are investigated to findsuitable models that define an overall accurate combustioninterpolation model.

The parameter θo in (28) defines the crank angle wherethe cylinder pressure starts to rise from the compressionasymptote. Hence, the difference between the estimatedvalue of θo and the injection angle is assumed to be theignition delay

θdelay = θo − αinj (30)

The ignition delay is modeled using an Arrheniustype formula as described in Chapter 10 of Heywood.6

Usually, the pre-exponential term, CA, contains a pressuredependence term.35 However here it is kept constantbecause with two parameters the agreement to thecalculated data is similar as if the pressure dependence termis included. The delay is written as

θdelay =180ωeng

π(CAe

EAT ) (31)

Figure 3. Ignition delay model in the top left, combustionduration model in the top right and efficiency model in thebottom left. Stars correspond to the separated fit for eachpressure measurement. Circles represent the proposedmodel. Each model fit contains the root-mean-square error(RMS) and the R-squared value. The errors of each of thethree models are plotted in the bottom right plots.

where CA, and the activation term, EA, are adjusted to thecalculated ignition delay values. The engine speed, ωeng ,is used to convert time to crank angle and T is the gastemperature when the fuel is injected. The model fit withonly two parameters is reasonable as can be seen in Figure3.

The combustion duration, represented by ∆θ is modeledas a linear function of the injected fuel mass flow.

∆θ = k∆θ,1 + k∆θ,2Wf (32)

This gives a good representation of the estimated values ascan be seen in Figure 3.

Defining the efficiency factor, ηf , as function of injectionangle is motivated by Figure 7.28 in Eriksson and Nielsen.3

Earlier injection results in higher peak pressures andtemperatures which consequently lead to more heat losses.On the other hand, later injection reduces the peak pressuresbut leads to higher expansion pressures and temperatureslater in the cycle. Thus, to capture this, the efficiency istaken as a linear function of the start of combustion, θo.

ηf = kηf ,1 + kηf ,2θo (33)

As can be seen in Figure 3 a linear model capturesreasonably well the estimated values from the separatedleast squares problems.

Finally, the parameter m is taken as a constant andinitially set as the mean value of the estimated values



to keep the model complexity low and avoid over-parameterization. In total, there are seven parameters to bedetermined for the cylinder pressure model.

3.7.6 Blowdown The blowdown phase starts at the EVOand finishes when the scavenging process begins at IPO.During the blowdown, the measured pressure decreasesfast to the lower pressure at IPO, and this decrease canbe approximated quite well with a linear model. Since wealready defined the cylinder pressure at IPO being equal topscav , the line connecting the two pressure points is entirelydefined, and it can be seen in Figure 2.

The temperature at EVO can be computed using thepressure asymptote, the trapped mass in the cylinder andthe ideal gas law. However, the temperature at IPO requiressome more assumptions if one wants to keep the modelanalytic and simple. When the exhaust valve is open, anenergy balance is applied to the open system. Followingthe reasoning in Section 5.5 of Cengel and Boles,28 auniform-flow process can represent reasonably well the realundergoing process. The primary assumption is that theenthalpy of the exiting gas is considered constant, and it iscomputed as the enthalpy of the gas at EVO, i.e., a throttlingprocess. This simplifies the model since an explicit equationfor T5 can be derived. The energy balance can be written as

m5cvT5 −m4cvT4 = −Ω4,5 − (m4 −m5)cpT4 (34)

where Ω is the work done by the gas to the piston from EVOto IPO. Since a linear pressure decrease from EVO to IPOhas been assumed, the work can be computed as

Ω4,5 =(p4 − p5)(V5 − V4)

2(35)

Using the ideal gas law to compute the mass inside thecylinder at IPO the explicit expression for T5 can be writtenas

T5 =p5V5cpT4

R

m4T4(cp − cv) + Ω4,5 + cvp5V5

R

(36)

where the dependencies of temperature and composition ofcv , cp and R have been dropped only in the notation. Thethermodynamic properties have been computed with thetemperature and concentration at EVO to be able to keep(36) explicit.

3.7.7 Cylinder Outlet Temperature Model Since differ-ent mass flows at different temperatures enter the exhaustmanifold during the cycle, a final balance is made to com-pute the resulting cylinder out temperature. This balanceis done assuming a perfect mixing process at constantpressure. Thus, the final expression is defined as

Tcyl =(m4 −m5)cpT4 +mshcpTscav + ηscavm5cpT5

(m4 −m5)cp +mshcp + ηscavm5cp(37)

where msh corresponds to the equivalent short-circuitedmass during one two-stroke cycle. The short-circuited massis computed as the delivered mass minus the trapped massplus the part of the cylinder mass that is not scavenged andremains in the cylinder. This is written as

msh = mdel −mtrap + (1− ηscav)m5 (38)

According to CFD analyses of the scavenging process,32,33

there is almost no short-circuit of pure scavenge air duringthe scavenging phase, and the mass that is not trapped inthe cylinder corresponds to a mix of fresh and burned gases.Nevertheless, to keep the model simple, it is assumed herethat the non-trapped gases are expelled at a temperatureequal to Tscav .

3.7.8 Cylinder Outlet Gas Composition The massfraction after the combustion is calculated assumingcomplete combustion of the injected fuel. Withoutincluding the nitrogen explicitly and considering leancombustion, the chemical reaction can be written as36

CHySz +(

1 +y

4+ z)O2 → CO2 +

y

2H2O + z SO2

(39)where y is the hydrogen to carbon ratio and z is the sulfurto carbon ratio in the fuel, which are known parameters.Knowing the trapped and fuel masses, the mass fractionafter the combustion, Xcomb, can be calculated in a similarway as in Nielsen et al.22 but getting masses insteadof moles as output. For the oxygen mass fraction, it iscomputed as

XO2,comb =mtrapXO2,in − (1 + y

4 + z)MO2

Mfmf

mtrap +mf(40)

For the rest of the gas concentrations, similar equations arederived based on (39). Finally, the mass fraction exitingthe cylinders, Xcyl can be computed by mixing the burnedgases mass fraction with the short-circuited scavengingmass fraction as follows.

Xcyl =Xcomb(mtrap +mf ) +Xscav (mdel −mtrap)

mdel +mf

(41)

3.8 Engine Brake PowerThe indicated work, Ωi, is computed as the integral of thepressure over the cylinder volume. The integration is doneanalytically for the polytropic compression and expansionprocesses, Ω1,2 and Ω3,4, for the blowdown, see (35), andfor the constant pressure between IPO and EVC, Ω5,1. Onthe other hand, for the combustion phase, Ω2,3, the integralis solved numerically. Since all cylinders are consideredto work under the same conditions, the indicated power iscomputed using the current engine speed and multiplying



the produced power by the number of engine cylinders,ncyl.

Pi =ωeng ncyl(Ω1,2 + Ω2,3 + Ω3,4 + Ω4,5 + Ω5,1)

2π(42)

The friction mean effective pressure, pmef , represents thefriction power losses normalized to the engine displacedvolume. These friction losses are modeled as a linearfunction of engine speed

pmef = kfr,1 + kfr,2 ωeng (43)

where parameters kfr,1 and kfr,2 are tuning parameters.Finally, the engine brake power, which is the effectivepower transferred to the propeller shaft, is calculated as

Pbrake = Pi −Vd ncyl ωeng

2πpmef (44)

where Vd is the displaced volume of one cylinder.

3.9 Cylinder Pressure Model vsMeasurements

Figure 4 contains several pressure diagrams for differentrelative loads and engine running modes to show that themeasured pressure can be predicted with the proposedmodel. The mean absolute relative errors for the indicatedpower, the peak cylinder pressure and the cylindercompression pressure at point 2 are; 1.49%, 1.48% and0.36% respectively.

3.10 Exhaust Manifold Temperature ModelHaving an accurate exhaust temperature model is importantfor the overall engine model accuracy. This is because theexhaust temperature together with the pressure sets theavailable power to the turbines, which in turn determinesthe rest of the model pressures. A heat transfer model isimplemented to account for the gas temperature loss fromthe cylinder outlet to the turbine inlets.

Initially, a model consisting of the heat transfer fromthe gas temperature to the ambient was considered. Instationary conditions, this model manages to get accuratepredictions of the measured temperatures. However, duringtransients, the temperature dynamics are very quick andreactive to changes in mass flow through the cylindersand to injected fuel. This makes the whole engine modeldynamics too fast compared to the measured signals.To slow down the model dynamics, the thermal inertiais increased by considering the wall temperature of theexhaust gas receiver as a model state. This slows down theexhaust temperature response since the thermal inertia ofthe wall is large.

The implemented heat transfer model is describedas Model 3 in Eriksson.37 Where the dynamic wall

Figure 4. Modeled cylinder pressure in solid lines comparedto the measured in dashed lines for different percentages ofthe maximum engine load and the corresponding enginerunning mode (ERM)

temperature becomes a model state. The dynamic equationfor the wall temperature is defined as

dTewdt

mewcew = Qi − Qe (45)

where Qi is the internal heat transfer from the exhaust gasto the manifold wall, calculated as

Qi = Wcyl cp(Tcyl − Tew)(1− e−hcv,iAew

Wcylcp ) (46)

where the convection coefficient, hcv,i, is considered as atuning parameter, and the exhaust wall area, Aew is fixed toits geometrical value. The heat transfer calculated in (46),is inserted in (4c) to get the exhaust manifold temperature.On the other hand, Qe is the external heat transfer from theexhaust manifold wall to the surroundings

Qe = Fv ε σAew(T 4ew − T 4

amb) + hcv,eAew(Tew − Tamb)(47)

The product of the gray body view factor and theemissivity, Fv · ε, is considered a tuning parameter. The



convection coefficient, hcv,e, is also a tuning parameter.The original model 337 considers conduction to the engineblock. However, since the engine block temperature is notmeasured, in the implementation this temperature is takenas the ambient temperature. In practice, this assumptionapproximates the conduction heat transfer lumped into theconvection, since the coefficient is estimated to fit theengine measurements.

3.11 Oxygen SensorThree different oxygen sensors measure the molar oxygenfraction of the gas in the scavenging manifold. A complexscavenging gas extraction system is installed in theengine, with the purpose to have a favorable pressure andtemperature for the oxygen sensors. This gas extractionsystem introduces uncertainty in the dynamic response ofthe oxygen sensor setup. The oxygen sensor response canbe approximated with a first order system with a puredelay22

d

dtO2,sen(t) =

O2,scav(t−∆tOs)− O2,sen(t)

τOs(48)

where O2,scav is the scavenging molar fraction and O2,sen

is the sensor reading. The time constants, τOs and ∆tOs,are both expected to be in the range of 10− 20s dependingon the scavenging pressure and gas extracting cloggingconditions.

3.12 Exhaust Temperature SensorThe dynamic response of the exhaust temperature sensor ismodeled as a first order system to compare the sensor signalto the modeled exhaust temperature.

d

dtTexh,sen(t) =

Texh(t)− Texh,sen(t)

τTs(49)

Texh,sen is the exhaust temperature with the sensordynamics, and Texh is the exhaust temperature state ofthe exhaust manifold, see Figure 1. The time constantτTs is unknown, but a value of 30 s gives a responsesimilar to the measured temperature. Section 5.2 containsfurther discussion about the agreement of the modeled andmeasured exhaust temperature signals.

4 Model ParameterizationThe proposed MVEM has three main internal feedbacksystems, the EGR loop, and the two turbochargers. Hence,the modeling errors of any subsystem will be coupledto the rest of the model and amplified. For instance,the errors in the exhaust temperature will affect theturbine power production which in turn will changethe scavenging pressure, and this finally will modify

the exhaust temperature prediction again. Due to this,balancing out the complete model, by readjusting the modelparameters is very important to obtain an overall accuratemodel.

In the following sections, the relative error is defined as

e[k] =ymod[k]− ymeas[k]

1/N∑Nj=1 ymeas[j]

(50)

where ymod is the model signal and ymeas corresponds tothe measured signal. Before computing the mean value,the absolute value of the relative error is taken to avoidcanceling out errors of different sign. Moreover, the relativeerrors in the text and tables are given in percentage,multiplying (50) by 100.

4.1 InitializationFirst of all, the parameters for the turbomachinery modelsare estimated and kept fixed. The turbocharger modelsare parameterized using the provided SAE maps. Thecompressor model is parameterized with the algorithmdescribed in Llamas and Eriksson.26 The EGR andauxiliary blowers are also parameterized using theperformance data.

During the parameterization of automotive MVEMs,the measured mass flow is an advantageous signal to beable to parameterize the models of the different enginecomponents by solving separated least squares problems.Unfortunately, this procedure cannot be followed here sincethe engine mass flow is not measured, which complicatesthe initialization of the model parameters. A way to remedythis issue is to use the compressor model together with themeasured speed and differential pressure to get an estimateof the engine mass flow when EGR is not running. Thisestimated mass flow is used to initialize the parameters ofthe valves, the flow restrictions and the coolers before thecomplete parameterization procedure.

The cylinder pressure model is initialized in Section3.7 to match the indicated power and cylinder pressure.In total seven parameters corresponding to (28), (31),(32) and (33). However, the cylinder model needs to beadjusted so that the cylinder output temperature agreeswith the measurements. Unfortunately, the cylinder outlettemperature is not measured. Hence, the cylinder modelis initialized together with the exhaust heat transfer modelto match the measured engine brake power and exhausttemperature. The initialization is started with the previouslymentioned seven parameters together with five morecorresponding to (46), (47) and (43). In order to preventthe parameterization algorithm from moving away from thedesigned cylinder pressure model described in Section 3.7,two additional signals are used in the objective functionsince the cylinder pressure measurements are not available



for the stationary points. These signals are the maximumpressure in the cylinder as well as the compression pressurein the cylinder. These pressures are measured by the enginecontrol system and used to control the combustion process.

4.2 Stationary ParameterizationThe procedure followed here to solve the parameterizationproblem is very similar to ones used in previous work.21

For each measured stationary point, the engine model issimulated in Simulink with the constant measured inputsignals. Once the model reaches stationarity, the modeloutput values are used to compute the relative errors withthe measured signals, using (50). This method involves alarge number of dynamic simulations per solver iterationwhich can make the complete process very computationallydemanding. However, for a given solver iteration, therequired dynamic simulations are independent of each otherso the process can be parallelized using the Matlab ParallelComputing Toolbox.

A least squares problem is then formed by stackingthe relative errors for each considered stationary pointand measured signal. The used solver is the MatlabOptimization Toolbox lsqnonlin, based on Gauss-Newton method with numerical approximation of thejacobian.

In the general case the cost function to be minimized iswritten as

Vstat(ϕ) =1

NS

S∑i=1

N∑n=1

(eirel[n])2 (51)

where N is the number of stationary points and S is thenumber of different signals involved in the estimation. Theestimated parameters are contained in a vector named ϕ.The parameters are subject to bound constraints to keep thesolver from choosing unreasonable values.

The complete parameterization is done in steps to ensurethat the solver is capable to solve the problem whileobtaining reasonable parameters. The steps start with theengine running mode that involves the lowest number ofparameters and increase in model complexity, the followingsubsections contain such steps and how they are performed.Iterating steps 3, 4 and 5 once, updating the current initialparameters with the previous step value has helped to finetune the model. Finally it is worth mentioning that theoptimization problem is highly nonlinear and non-convex,so there is no guarantee that the found solution is the globaloptimum. In order to ensure that the parameters are a goodsuitable solution, the proposed steps have been started withdifferent initial values and the obtained parameters havebeen compared and the best ones are kept. The parameterselection criterion is to check which set of parameters givesthe lowest sum of squared residuals.

Step 1 ERM 2 stationary points. Only the mainturbocharger is working and there is no EGR operation. Therelative errors used in (51) are computed with the measuredsignals: pscav , pc,1, ωtc,1, Pbrake, and Texh. The estimatedparameters are

ϕ = [Aac,1, AAux, AEGB , Acyl, Aout, Ain, ηmech,1]

Step 2 Stationary points with ERM 1 and 2. Bothturbochargers are working, but there is no EGR operation.The parameters estimated in the previous step are used asstarting point. The relative errors used in (51) are computedwith the measured signals: pscav , pc,1, ωtc,1, ωtc,2, Pbrakeand Texh. The estimated parameters are

ϕ = [Aac,1, Aac,2, AAux, AEGB , Acyl,

Aout, Ain, ηmech,1, ηmech,2]

Step 3 Stationary points with ERM 3 and 4, EGR is nowrunning. While keeping the rest of the parameters fixed, theEGR loop flow restrictions are estimated. The relative errorsused in (51) are computed with the measured signals: pscav ,pc,1, ωtc,1, Pbrake, Texh, XO2,scav

, peb,in and peb,out. Theestimated parameters are

ϕ = [ASDV , Aac,egr, ACOV ]

Step 4 Using all stationary points the exhaust manifoldheat transfer model is re-parameterized. All other param-eters are kept constant. The relative errors used in (51)are computed with the measured signals: pscav , pc,1, ωtc,1,ωtc,2, Pbrake, Texh and XO2,scav . This step involves threeheat transfer coefficients of the exhaust manifold tempera-ture model

ϕ = [hcv,i, hcv,e, Fv · ε]

Step 5 Using all stationary points the restriction parame-ters are re-parameterized starting with the parameter valuesobtained in the previous steps. The relative errors usedin (51) are computed with the measured signals: pscav ,pc,1, ωtc,1, Pbrake, Texh, XO2,scav , peb,in and peb,out. Theestimated parameters are

ϕ = [Aac,1, Aac,2, AAux, AEGB , Acyl,

Aout, Ain, ηmech,1, ηmech,2]

4.3 Dynamic ParameterizationThis final parameterization step involves estimating theparameters that define the dynamic behavior of the model,which corresponds to the turbocharger inertias and thecontrol volume sizes.

A least squares problem that minimizes the dynamicdeviation of the measured states can be solved to estimate



the model dynamic parameters.19–21 However, solvingthis optimization can be complex and it requires suitabletransient datasets. Instead, this procedure is simplified hereby setting the value of the control volumes to reasonablevalues of the real pipe volume sizes based on the enginedesign drawings. The turbocharger inertias are set to thenominal values included in the SAE maps. This is agood strategy to define the model dynamic parametersin case only stationary shop test data is available forparameterization.

The model response with the previously mentioneddynamic parameters is quite good. However, when EGRis started or stopped the model response is faster thanthe measured. This issue is solved by increasing themain turbocharger inertia by 30% of the nominal SAEvalue, which improves the dynamic response of the model.This behavior was also noticed in previous versions ofthe model,20,21 where the exhaust heat transfer was notmodeled, and the turbocharger inertia had to be increasedsubstantially more. Sections 5.2 and 6 contain furtherdiscussion about this issue.

5 Model ValidationThe model is validated by comparing the measured signalsto the model predictions. First, this is done evaluating themodel in stationary conditions. Second, the model is alsovalidated under transient conditions.

5.1 Stationary ResultsThe absolute relative stationary errors are presented inTable 1 for both the parameterization and validation data.The mean relative errors for the tuning data are in generalunder 3.35%, which is a good indication that the modelis capable of predicting the main engine outputs for anyload and ERM value. It is worth to mention that the highesterrors are when EGR is active, i.e., in ERM 3 and 4. Thisis expected since the engine becomes more complex whenthe EGR system is active, more engine components areworking, and thus more uncertainty is added to the system.Note also that Table 1 contains the number of stationarypoints available for each ERM, and the modes 2 and 3 havefewer points than modes 1 and 4 to tune and to compute theerrors of the model. Only points with EGR active have beenused to compute the molar oxygen concentration error, aswell as only points operating at ERM number 1 are used inthe second turbocharger speed error calculation.

The Figure 5 depicts the model predictions vs. themeasured values for the parameterization data. The overallaccuracy of the model is good, from 10% to almost 90%loads, with and without EGR operation.

In order to show that the model can achieve accuratepredictions not only with the points used in the

Figure 5. Stationary model predictions vs. measurements fordifferent engine signals using the parameterization stationarydata. Note that the molar oxygen concentration has threedifferent sensors in the same location.

parameterization, the model predictions are compared fora set of stationary points not used in the parameterization.These stationary points are named validation points, andtheir numerical errors are also given in Table 1, andshown graphically in Figure 6. This set of data pointsare close to the previous parameterization data since theyare also extracted from normal operation of the engineduring sailing. However, they are not the same operatingpoints. Note that the errors for the ERM 3 are unfortunatelycomputed with a single measured point. This is of coursenot desirable, but the engine is very seldom operating in thisconditions. The errors increase only slightly when goingfrom tuning data to validation data, as seen in Table 1,which is a good indication that the model will be ableto predict the engine operation accurately. Table 1 alsocontains the root-mean-square errors (RMS) for each of thelisted measured signals, which indicates the spread of themodel errors. The RMS values for estimation and validationdata are similar, which is also an indication of good modelaccuracy.



Table 1. Absolute relative errors [%] of the model for both estimation and validation data, separated for different ERMs. Thetable also contains the combined root-mean-square error (RMS) of all listed residuals.

Estimation Stationary Points Validation Stationary PointsERM 1 2 3 4 RMS 1 2 3 4 RMSPoints 19 9 4 20 52 17 9 1 13 40pscav 1.30 2.53 2.21 1.87 4.66 · 103 Pa 1.94 2.34 3.07 1.64 4.93 · 103 Paωtc,1 1.35 2.99 2.88 2.95 23.26 rad/s 1.35 2.59 0.73 2.54 20.57 rad/sωtc,2 1.14 − − − 19.04 rad/s 1.24 − − − 21.73 rad/spc,1 1.98 3.16 2.56 1.78 5.30 · 103 Pa 2.78 3.16 4.75 1.42 5.88 · 103 PaTexh 0.97 2.06 2.80 1.65 12.22 K 0.77 2.65 4.15 1.30 12.28 KPbrake 1.66 1.61 2.40 1.94 1.93 · 105 W 2.33 1.71 4.35 1.53 2.28 · 105 W

XO2,scav − − 1.54 1.69 0.34 % − − 1.40 2.19 0.46 %peb,in − − 2.91 2.90 5.33 · 103 Pa − − 4.10 2.59 4.75 · 103 Papeb,out − − 2.76 3.35 5.38 · 103 Pa − − 4.20 1.98 5.67 · 103 Pa

Figure 6. Stationary model predictions vs. measurements fordifferent engine signals using the validation stationary data.Note that the molar oxygen concentration has three differentsensors in the same location.

5.2 Dynamic Results

Four different dynamic scenarios are simulated andcompared to the model predictions to validate the modeldynamics. The four scenarios are load increase, loaddecrease, EGR start and EGR stop. All scenarios are

focused on low load operation, which is the most uncertainoperating area of the model since the auxiliary blower isrunning and the turbochargers are operating outside of themapped area. Moreover, most of the maneuvering transientsoccur at low loads.

The first scenario with increasing load and thetwo turbochargers running is depicted in Figure 7.Approximately the engine load goes from 20 % to 35 %with the auxiliary blowers running. The model shows agood agreement with the measured data for both pressuresignals. For the turbocharger speed signals the agreementis also good, however, there is a higher stationary error.The fast temperature sensor is plotted together with theexhaust temperature in dashed line and the output of thetemperature sensor described in (49). As can be seen,the modeled sensor temperature agrees quite well to themeasurement, while the modeled exhaust temperature isfaster. A time constant of 30 s has been found to be areasonable approximation of the sensor dynamics for allthe cases tested. In reality, the sensor might be faster andthe real exhaust temperature slower. Unfortunately, this isnot possible to determine with certainty with the currentlyavailable measurements.

The next scenario, shown in Figure 8, is an engineload decrease. Stationary errors are higher than in theprevious case, but the dynamic response of the modelcompared to pressures and turbocharger speeds is alsogood. This dynamic dataset has the particularity to startthe Auxiliary blower approximately at 450 s. As can beseen in Figure 8, the model is capable of capturing theeffect that the auxiliary blower has on the turbochargerspeeds and the pressures. The auxiliary blowers starttime constant, described in (14), is adjusted to matchthe dynamic response. This, as discussed in Section 3.5,represents the time it takes for the blowers to accelerate tothe working speed.

The third scenario consists of an EGR start and it isdepicted in Figure 9. The engine load is constant andclose to 20 % of the maximum rated power, the auxiliary



Figure 7. Model transient predictions and measured signalsfor a Load increase scenario. The right axes in the bottom plotcorrespond to the percentage of engine load, depicted with adotted line.

blower is active during the transient. The EGR valveopens approximately at 80 s and the oxygen level inthe manifold starts to drop to later follow the orderedEGR blower speed fluctuation. The agreement betweenmeasured and modeled oxygen volume fractions is good.Note that the modeled oxygen is passed through the sensordynamics model described in (48), to have both signalssynchronized. The modeled temperature drops faster thanthe measurement when EGR starts to flow. However, whenthe sensor dynamics are included, the temperature responseagrees much more to the measured signal.

The last scenario, shown in Figure 10 is an EGR stopat 25% engine load with the auxiliary blower enabled. Thepressure and turbocharger speed dynamics are good, despite

Figure 8. Model transient predictions and measured signalsfor a Load decrease scenario followed by an auxiliary blowerstart. The right axes in the bottom plot correspond to thepercentage of engine load, depicted with a dotted line.

the latter having a larger stationary error. The temperatureresponse is also faster than the measured, and peaks ata higher value. This introduces a longer transient untilthe modeled temperature settles at the stationary value,compared to the effects observed in the measured signal.The oxygen level goes back to atmospheric conditions afterthe EGR valve angle is closed, the agreement between themodel and the measured signals is good.

6 Guidelines for Future EngineMeasurements

Gathering sufficient measurement data at different operat-ing points takes a lot of time since the measurement data isrecorded while the ship is sailing. Hence, having guidelines



Figure 9. Model transient predictions and measured signalsfor an EGR start scenario. Note that the three measuredOxygen molar concentrations are plotted as dashed lines. Inthe same plot, the EGR blower speeds are shown as dottedlines with the corresponding axes on the right side.

of what measurements are important beforehand could beuseful when parameterizing this model to other engines,and also to further improvements of the proposed enginemodel. The experience gained during the development ofthis engine model is summarized in this section.

First of all, fast temperature sensors mounted close tothe cylinder exhaust valves would be very useful. Thosemeasurements would help to validate the assumptionstaken for the cylinder out temperature model described inSection 3.7. Moreover, those measurements together withfast measurements of the exhaust wall and the engine roomtemperature would permit to decouple the parameterizationof the cylinder temperature model from the heat transfermodel of the exhaust manifold.

Figure 10. Model transient predictions and measured signalsfor an EGR stop scenario. Note that the three measuredOxygen molar concentrations are plotted as dashed lines. Inthe same plot, the EGR blower speeds are shown as dottedlines with the corresponding axes on the right side.

Regarding the auxiliary blower, the engine designerdoes not choose of the specific blower that will beinstalled. Hence, the only information available are thetwo operating specifications that have to be fulfilled by theblower chosen by the engine manufacturer. Placing pressuresensors and the inlet and outlet of the blower as well asa blower speed measurement would be helpful for lowload parameterization. In addition, it would be useful forimproving the proposed dynamic model (14), by taking intoconsideration the speed dynamics of the blower.

Mass flow measurements could also be very helpful tovalidate the model extrapolation at low loads when theauxiliary blower is running, and the compressor is workingwell outside the mapped area. This measurement would



be also helpful for validating the choice of a compressiblerestriction as the cylinder mass flow model. Furthermore,the parameterization process could be simplified, since thedifferent submodels could be decoupled. Unfortunately, theinstallation of pressure based mass flow sensors is quitecomplicated since usually, they introduce too high-pressuredrops.

7 Propeller and Ship Resistance ModelsWith the purpose to simulate the complete ship propulsionsystem, models for the propeller and the ship resistanceare implemented. These models define two more states,the engine shaft rotational speed and the ship sailingspeed. Moreover, they also provide the resistive load thatthe previously described two-stroke engine model hasto overcome. The modeling follows the same steps aspresented in Theotokatos and Tzelepis,38 which in turnsfollows the well-known modeling approach developed inHoltrop and Mennen.39

7.1 Propeller and Shaft ModelsThe ship propeller required torque, QP , and providedthrust, TP , are computed using the definition of the non-dimensional coefficients of torque and thrust,40 KQ andKT respectively.

QP = KQ ρsw N2P D

5P (52)

TP = KT ρsw N2P D

4P (53)

The non-dimensional coefficients are computed withthe help of the Wageningen B-Screw Series polynomialsdescribed in Van Lammeren et al.41 These polynomials arefunction of the propeller geometrical parameters, availablefrom the propeller technical specifications, and the advancespeed, JA, which is computed as

JA =vA

DP nP(54)

where vA is the advance speed of the propeller relativeto the water, DP is the propeller diameter, and nP is thepropeller speed in revolutions per second.

The shaft speed is calculated with the followingdifferential equation

dωengdt

=Qbrake −QP

Jeng + Jshaft + JP + Jent(55)

where Qbrake is the engine brake torque, and the differentshaft system inertias are denoted as J in the denominator.The entrained water inertia, Jent, is calculated using theregression model described in Parsons42 for WageningenB-Series propellers.

7.2 Ship Resistance ModelWith the produced propeller thrust and the calculated shipresistance, the current speed of the ship can be calculated as

dvshipdt

=TP (1− t)−Rshipmship +mhydro

(56)

where t is the thrust deduction coefficient, Rship the shipresistance, mship is the mass of the ship calculated withthe displaced water volume and density. The added mass,mhydro, represents the hydrodynamic inertial effects of thesurrounding water. It is calculated using the linear equationproposed in Oltmann.43 The resistance and the thrustdeduction coefficient are computed following the statisticalprediction method described in Holtrop and Mennen,39

with the updated equations presented in Holtrop.44 Thespecific container ship geometrical characteristics are usedin the calculation of Rship.

The dynamic model formed by (55) and (56), isimplemented in Simulink. Since not all hull designparameters required for calculating the ship resistanceare known, the unknown values are adjusted, so themodel agrees with the measurements. The measured enginebraking power is used as input, and the calculated shaftrotational speed and vessel speed are plotted against themeasured signals in Figure 11. As it can be seen, the modelgives a reasonable estimate of the engine loading profile,while the vessel speed prediction is worse in general. It isimportant to mention that the model prediction could beimproved by knowing more about the current loading of thevessel for each stationary point. Parameters like the draughtand displacement depend on the amount of cargo and hereare taken as constants to have an approximation for all thepoints shown in Figure 11. Moreover, the ship resistance ishighly dependent on the sea conditions and the hull foulinglevel,45 and these conditions are unknown for the collectedstationary points. Despite all these unknown parameters,this is a suitable starting point to simulate the completecontainer ship during transients. Moreover, it also enablesto carry out simulations to study the effects of increasedresistance based on sea conditions.38

7.3 Engine Speed GovernorThe engine speed governor controls the fuel index value,Y , to match an ordered engine speed value. The governoris modeled as a PI controller,10 which in the frequencydomain it is written as

Y = Kp ∆Neng +Ki ∆Neng

s(57)

where ∆Neng is the difference between the desired enginespeed and the current engine speed. The parameters,Kp andKi are the proportional and integral gains respectively.



Figure 11. Propeller and ship resistance models vsmeasurements for the measured stationary operating points.

Figure 12 shows a simulation of the engine modeltogether with the ship resistance model and the imple-mented engine speed governor. The ordered speed value isnot available in the measurements, so in order to comparethe ship speed dynamics, the measured engine speed is usedas the governor input. As can be seen in the figure the modelcan predict reasonably well the measured necessary enginebrake power based on the ordered fuel index. Note that adelay in the measured brake power can be observed, thisis due to a delay in the measured engine torque signal. Ascan be seen in Figure 12 the measured engine speed and fuelindex are synchronized with the model signals. On the otherhand, the vessel speed prediction has a similar dynamicbehavior as the measured signal despite the larger stationarydifference. This difference is a consequence of the propellerand resistance model errors, as discussed in the previoussection. The complete vessel model is simulated in MatlabSimulink, in a standard desktop computer, the simulationshown in Figure 12 is performed 55 times faster than realtime.

8 ConclusionsAn MVEM of a large two-stroke diesel engine with EGRis described and validated against measured data of a realengine operating in a sailing container ship. The requiredsimulation time in a standard desktop computer is in general50 times faster than real time. This fast simulation timemakes the model suitable for the intended application ofEGR and fuel controller development. A new analyticcylinder pressure model is also developed using cylinderpressure measurements from the real engine. The cylindermodel is shown to be able to capture the effects of differentcontrol inputs to the prediction of engine power output andcylinder pressure.

The stationary parameterization procedure, necessarydue to the low measurement data availability, managesto adjust the proposed MVEM response to the measuredsignals. The model is shown to capture the stationary engineoperation for a wide span of engine loads well, from

Figure 12. Comparison between the dynamic signals of thecomplete ship model with the speed governor and themeasured signals.

10% to 90%, with and without the EGR system activated.The stationary relative errors are in general under 3.35%for both estimation and validation data which is a goodindication of model accuracy.

To simplify the dynamic parameterization, the dynamicparameters are set to the nominal geometric values. Thedynamic validation of the model shows that it is capableof following the measured engine signals during transients.However, the model transient response is for some casesfaster than the measured, especially during the transientsintroduced when EGR is started and stopped. The appliedsolution is to increase the main turbocharger inertia by 30%.

The use of control-oriented simulation models fordesigning and benchmarking EGR and fuel controllerswill increase in the near future. Thus, the lessons learnedduring the complete model development project aresummarized here for the research community. Essentialsignals to be included in future measurement campaignsand measurement setups for newly built engines areidentified. These signals will improve the parameterizationprocess of the proposed model and also help to validatefurther the model assumptions.



Finally, models for the propeller and ship resistance arealso investigated together with an engine speed governor.It is shown that these models can be used together withthe proposed MVEM to simulate the complete propulsionsystem of container ships during maneuvering. This is ofparticular interest since the analysis of EGR controllerperformance during loading transients is one of the mostimportant scenarios where the EGR control has to beimproved. The main reason for this is that the more strictNOx emission limits are enforced in coastal areas wherethe ships need to be able to maneuver when approachingharbors.

Acknowledgements

MAN Diesel & Turbo is greatly acknowledged for their supportand discussions about the engine and for providing measurementdata.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This project has received funding from the European Union’sHorizon 2020 research and innovation programme under grantagreement No 634135.

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Appendix I

NomenclatureQ Heat transfer

O2 Oxygen Molar Fraction

X Molar Fraction

A Area or Equivalent Area

D Diameter

J Inertia or propeller advance speed

M Molar mass

N Rotational Speed in [rpm]

P Power

Q Torque

R Gas Constant or Resistance force

RH Relative Humidity

T Temperature or Thrust

V Volume

W Mass Flow

X Mass Fraction



Y Fuel Index

CFD Computational Fluid Dynamics

EGR Exhaust Gas Recirculation

ERM Engine Running Mode

EV C Exhaust Valve Closing

EV O Exhaust Valve Opening

IPC Intake Port Closing

IPO Intake Port Opening

MVEM Mean Value Engine Model

NECA NOx Emission Control Area

RMS Root-mean-square error

cp Specific heat at constant pressure

cv Specific heat at constant volume

e Relative error

h Heat Transfer coefficient

k Model parameter

m Mass

n Rotational Speed in [rps]

p Pressure

t Thrust deduction coefficient

u Control input

v Velocity

Greek Symbols

α Crank Angle (control input)

ε Emissivity

η Efficiency

γ Specific heats ratio

Ω Work

ω Rotational Speed in [rad/s]

Φ Flow coefficient

Π Pressure ratio

Ψ Energy Transfer coefficient

ρ Density

σ Boltzmann constant

θ Crank Angle

ϕ Vector of parameters

Subscripts

A Advance

ac Air Cooler

amb Ambient

Aux Auxiliary Blower

c Compressor

co Compression

comb Combustion

COV Cut-Out Valve

cv Convection

cyl Cylinder

del Delivered

e External

eb EGR Blower

eff Effective

EGB Exhaust Gas Bypass

eng Engine

ent Entrained water

ew Exhaust wall

ex Expansion

exh Exhaust

f Fuel

i Internal

in Inlet

inj Injection

meas Measured

mech Mechanical

mix Gas Mixer

mod Model

out Outlet

P Propeller

scav Scavenging

SDV Shut Down Valve

sen Sensor

shaft Engine crankshaft

ship Ship

sw Sea water

t Turbine

tc Turbocharger

v Valve

Appendix II: Derivatives ofthermodynamical propertiesThe working fluid is assumed to be a thermally perfectgas formed by a mixture of different gases. Hence,when deriving the expressions for the control volumedynamics described in Section 3.2, partial derivatives of thethermodynamical properties appear. This Appendix briefly



introduces how these expressions are computed based onthe NASA polynomials25 for the considered expressions.

The molar mass and the gas constant of a mixture of Nspecies is computed as in Cengel and Boles28

M(X) =

∑Nj=1Xj∑N

j=1Xj/Mj

(58)

R(X) =R

M(X)(59)

The derivative of the gas constant with respect to the massfractions is computed as

dR(X)

dX=

−RM(X)2

∇xM(X) = −R

...

1M(X) −

1Mi

...

(60)

It is possible to transform from molar to mass fractions andvice-versa using the following equations

Xi =XiMi

MXi =

XiM

Mi(61)

for a given gas i in the mixture.With the definition of the NASA polynomials, the

specific heat at constant pressure of gas i in the mixtureis computed as

cp,i(T ) =R

Mi(ai,1 + ai,2T + ai,3T

2 + ai,4T3 + ai,5T

4)︸︷︷︸Pn,i(T )

(62)where ai,x are the coefficients of the NASA polynomial,25

Pn,i(T ), for a given gas in the mixture, i. Considering thegas as a nonreacting mixture, the specific heat at constantpressure can be computed as the sum of the contributionsof each mixture component28

cp(X,T ) =

N∑j=1

Xjcp,j(T ) (63)

The specific heat at constant volume can be computedfrom the mixture gas constant and the specific heat atconstant pressure as follows since an ideal gas is assumed28

cv(X,T ) = cp(X,T )−R(X) (64)

The partial derivatives of the specific heat at constantvolume appear when differentiating the energy equation in(4a). They can be computed from the previous equationusing the chain rule

∂cv(X,T )

∂X=∂cp(X,T )

∂X− dR(X)

dX(65)

∂cv(X,T )

∂T=∂cp(X,T )

∂T(66)

The required partial derivatives of the specific heat atconstant pressure can be computed as

∂cp(X,T )

∂X= ∇xcp(X,T ) =

...

RMiPn,i(T )

...

(67)

∂cp(X,T )

∂T=

N∑j=1

XjR

Mj

dPn,i(T )

dT(68)

where dPn,i(T )dT is the derivative with respect to the control

volume temperature of the NASA polynomial of thearbitrary gas i in the mixture.


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