A two-step model order reduction method is investigated in order to overcome problems of certain one-step methods. Not only optimal combinations of one-step reductions are considered but also the selection of a suitable intermediate dimension (ID) is described. Several automated selection methods are presented and their application tested on a gear box model. The implementation is realized using a Matlab-based Software MORPACK. Several recommendations are given towards the selection of a suitable ID, and problems in Model Order Reduction (MOR) combinations are pointed out. A pseudo two-step is suggested to reduce the full system without any modal information. A new node selection approach is proposed to enhance the SEREP approximation of the system’s response for small reduced representations.:Contents Kurzfassung..........................................................................................iv Abstract.................................................................................................iv Nomenclature........................................................................................ix 1 Introduction........................................................................................1 1.1 Motivation........................................................................................1 1.2 Objectives........................................................................................1 1.3 Outline of the Thesis........................................................................2 2 Theoretical Background.......................................................................3 2.1 Finite Element Method......................................................................3 2.1.1 Modal Analysis...............................................................................4 2.1.2 Frequency Response Function.......................................................4 2.2 Model Order Reduction.....................................................................5 2.3 Physical Subspace Reduction Methods.............................................7 2.3.1 Guyan Reduction...........................................................................7 2.3.2 Improved Reduced System Method...............................................8 2.4 Modal Subspace Reduction Methods...............................................10 2.4.1 Modal Reduction...........................................................................11 2.4.2 Exact Modal Reduction..................................................................11 2.4.3 System Equivalent Reduction Expansion Process.........................13 2.5 Krylov Subspace Reduction Methods...............................................14 2.6 Hybrid Subspace Reduction Methods..............................................15 2.6.1 Component Mode Synthesis........................................................16 2.6.2 Hybrid Exact Modal Reduction......................................................19 2.7 Model Correlation Methods.............................................................21 2.7.1 Normalized Relative Frequency Difference...................................21 2.7.2 Modified Modal Assurance Criterion.............................................22 2.7.3 Pseudo-Orthogonality Check.......................................................22 2.7.4 Comparison of Frequency Response Function.............................23 3 Selection of Active Degrees of Freedom............................................25 3.1 Non-Iterative Methods...................................................................26 3.1.1 Modal Kinetic Energy and Variants..............................................26 3.1.2 Driving Point Residue and Variants..............................................27 3.1.3 Eigenvector Component Product..................................................28 3.2 Iterative Reduction Methods...........................................................29 3.2.1 Effective Independence Distribution.............................................29 3.2.2 Mass-Weighted Effective Independence.......................................32 3.2.3 Variance Based Selection Method.................................................33 3.2.4 Singular Value Decomposition Based Selection Method................34 3.2.5 Stiffness-to-Mass Ratio Selection Method.....................................34 3.3 Iterative Expansion Methods...........................................................35 3.3.1 Modal-Geometrical Selection Criterion...........................................36 3.3.2 Triaxial Effective Independence Expansion...................................36 3.4 Measure of Goodness for Selected Active Set..................................39 3.4.1 Determinant and Rank of the Fisher Information Matrix................39 3.4.2 Condition Number of the Partitioned Modal Matrix........................40 3.4.3 Measured Energy per Mode..........................................................40 3.4.4 Root Mean Square Error of Pseudo-Orthogonality Check.............41 3.4.5 Eigenvalue Comparison................................................................41 4 Two-Step Reduction in MORPACK.......................................................42 4.1 Structure of MORPACK.....................................................................42 4.2 Selection of an Intermediate Dimension.........................................43 4.2.1 Intermediate Dimension Requirements........................................44 4.2.2 Implemented Selection Methods..................................................45 4.2.3 Recommended Selection of an Intermediate Dimension...............48 4.3 Combination of Reduction Methods.................................................49 4.3.1 Overview of All Candidates..........................................................50 4.3.2 Combinations with Modal Information.........................................54 4.3.3 Combinations without Modal Information....................................54 5 Applications........................................................................................57 5.1 Gear Box Model...............................................................................57 5.2 Selection of Additional Active Nodes................................................58 5.3 Optimal Intermediate Dimension......................................................64 5.4 Two-Step Model Order Reduction Results........................................66 5.5 Comparison to One-Step Model Order Reduction Methods..............70 5.6 Comparison to One-Step Hybrid Model Order Reduction Methods...72 5.7 Proposal of a New Approach for Additional Node Selection..............73 6 Summary and Conclusions...................................................................77 7 Zusammenfassung und Ausblick..........................................................79 Bibliography............................................................................................81 List of Tables..........................................................................................86 List of Figures.........................................................................................88 A Appendix.............................................................................................89 A.1 Results of Two-Step Model Order Reduction.....................................89 A.2 Data CD............................................................................................96 / Mehrschrittverfahren der Modellreduktion werden untersucht, um spezielle Probleme konventioneller Einschrittverfahren zu lösen. Eine optimale Kombination von strukturmechanischen Reduktionsverfahren und die Auswahl einer geeigneten Zwischendimension wird untersucht. Dafür werden automatische Verfahren in Matlab implementiert, in die Software MORPACK integriert und anhand des Finite Elemente Modells eines Getriebegehäuses ausgewertet. Zur Auswahl der Zwischendimension werden Empfehlungen genannt und auf Probleme bei der Kombinationen bestimmter Reduktionsverfahren hingewiesen. Ein Pseudo- Zweischrittverfahren wird vorgestellt, welches eine Reduktion ohne Kenntnis der modalen Größen bei ähnlicher Genauigkeit im Vergleich zu modalen Unterraumverfahren durchführt. Für kleine Reduktionsdimensionen wird ein Knotenauswahlverfahren vorgeschlagen, um die Approximation des Frequenzganges durch die System Equivalent Reduction Expansion Process (SEREP)-Reduktion zu verbessern.:Contents Kurzfassung..........................................................................................iv Abstract.................................................................................................iv Nomenclature........................................................................................ix 1 Introduction........................................................................................1 1.1 Motivation........................................................................................1 1.2 Objectives........................................................................................1 1.3 Outline of the Thesis........................................................................2 2 Theoretical Background.......................................................................3 2.1 Finite Element Method......................................................................3 2.1.1 Modal Analysis...............................................................................4 2.1.2 Frequency Response Function.......................................................4 2.2 Model Order Reduction.....................................................................5 2.3 Physical Subspace Reduction Methods.............................................7 2.3.1 Guyan Reduction...........................................................................7 2.3.2 Improved Reduced System Method...............................................8 2.4 Modal Subspace Reduction Methods...............................................10 2.4.1 Modal Reduction...........................................................................11 2.4.2 Exact Modal Reduction..................................................................11 2.4.3 System Equivalent Reduction Expansion Process.........................13 2.5 Krylov Subspace Reduction Methods...............................................14 2.6 Hybrid Subspace Reduction Methods..............................................15 2.6.1 Component Mode Synthesis........................................................16 2.6.2 Hybrid Exact Modal Reduction......................................................19 2.7 Model Correlation Methods.............................................................21 2.7.1 Normalized Relative Frequency Difference...................................21 2.7.2 Modified Modal Assurance Criterion.............................................22 2.7.3 Pseudo-Orthogonality Check.......................................................22 2.7.4 Comparison of Frequency Response Function.............................23 3 Selection of Active Degrees of Freedom............................................25 3.1 Non-Iterative Methods...................................................................26 3.1.1 Modal Kinetic Energy and Variants..............................................26 3.1.2 Driving Point Residue and Variants..............................................27 3.1.3 Eigenvector Component Product..................................................28 3.2 Iterative Reduction Methods...........................................................29 3.2.1 Effective Independence Distribution.............................................29 3.2.2 Mass-Weighted Effective Independence.......................................32 3.2.3 Variance Based Selection Method.................................................33 3.2.4 Singular Value Decomposition Based Selection Method................34 3.2.5 Stiffness-to-Mass Ratio Selection Method.....................................34 3.3 Iterative Expansion Methods...........................................................35 3.3.1 Modal-Geometrical Selection Criterion...........................................36 3.3.2 Triaxial Effective Independence Expansion...................................36 3.4 Measure of Goodness for Selected Active Set..................................39 3.4.1 Determinant and Rank of the Fisher Information Matrix................39 3.4.2 Condition Number of the Partitioned Modal Matrix........................40 3.4.3 Measured Energy per Mode..........................................................40 3.4.4 Root Mean Square Error of Pseudo-Orthogonality Check.............41 3.4.5 Eigenvalue Comparison................................................................41 4 Two-Step Reduction in MORPACK.......................................................42 4.1 Structure of MORPACK.....................................................................42 4.2 Selection of an Intermediate Dimension.........................................43 4.2.1 Intermediate Dimension Requirements........................................44 4.2.2 Implemented Selection Methods..................................................45 4.2.3 Recommended Selection of an Intermediate Dimension...............48 4.3 Combination of Reduction Methods.................................................49 4.3.1 Overview of All Candidates..........................................................50 4.3.2 Combinations with Modal Information.........................................54 4.3.3 Combinations without Modal Information....................................54 5 Applications........................................................................................57 5.1 Gear Box Model...............................................................................57 5.2 Selection of Additional Active Nodes................................................58 5.3 Optimal Intermediate Dimension......................................................64 5.4 Two-Step Model Order Reduction Results........................................66 5.5 Comparison to One-Step Model Order Reduction Methods..............70 5.6 Comparison to One-Step Hybrid Model Order Reduction Methods...72 5.7 Proposal of a New Approach for Additional Node Selection..............73 6 Summary and Conclusions...................................................................77 7 Zusammenfassung und Ausblick..........................................................79 Bibliography............................................................................................81 List of Tables..........................................................................................86 List of Figures.........................................................................................88 A Appendix.............................................................................................89 A.1 Results of Two-Step Model Order Reduction.....................................89 A.2 Data CD............................................................................................96

Die FEM-MKS-Kopplung erfordert Modellordnungsreduktions-Verfahren, die mit kleiner reduzierter Systemdimension das Übertragungsverhalten mechanischer Strukturen abbilden. Rationale Krylov-Unterraum-Verfahren, basierend auf dem Arnoldi-Algorithmen, ermöglichen solche Abbildungen in frei wählbaren, breiten Frequenzbereichen. Ziel ist der Entwurf einer fehlerüberwachten Modelreduktion auf Basis von Krylov-Unterraumverfahren und Anwendung auf ein strukturmechanisches Model. Auf Grundlage der Software MORPACK wird eine Arnoldi-Funktion erster Ordnung um interpolativen Startvektor, Eliminierung der Starrkörperbewegung und Reorthogonalisierung erweitert. Diese Operationen beinhaltend, wird ein rationales, interpolatives SOAR-Verfahren entwickelt. Ein rationales Block-SOAR-Verfahren erweist sich im Vergleich als unterlegen. Es wird interpolative Gleichwichtung verwendet. Das Arnoldi-Verfahren zeichnet kleiner Berechnungsaufwand aus. Das rationale, interpolative SOAR liefert kleinere reduzierte Systemdimensionen für gleichen abgebildeten Frequenzbereich. Die Funktionen werden auf Rahmen-, Getriebegehäuse- und Treibsatzwellen-Modelle angewendet. Zur Fehlerbewertung wird eigenfrequenzbasiert ein H2-Integrationsbereich festgelegt und der übertragungsfunktionsbasierte, relative H2-Fehler berechnet. Es werden zur Lösung linearer Gleichungssysteme mit Matlab entsprechende Löser-Funktionen, auf Permutation und Faktorisierung basierend, implementiert.:1. Einleitung 1.1. Motivation 1.2. Einordnung 1.3. Aufbau der Arbeit 2. Theorie 2.1. Simulationsmethoden 2.1.1. Finite Elemente Methode 2.1.2. Mehrkörpersimulation 2.1.3. Kopplung der Simulationsmethoden 2.2. Zustandsraumdarstellung und Reduktion 2.3. Krylov Unterraum Methoden 2.4. Arnoldi-Algorithmen erster Ordnung 2.5. Arnoldi-Algorithmen zweiter Ordnung 2.6. Korrelationskriterien 2.6.1. Eigenfrequenzbezogene Kriterien 2.6.2. Eigenvektorbezogene Kriterien 2.6.3. Übertragungsfunktionsbezogene Kriterien 2.6.4. Fehlerbewertung 2.6.5. Anwendung auf Systeme sehr großer Dimension 3. Numerik linearer Gleichungssysteme 3.1. Grundlagen 3.2. Singularität der Koeffizientenmatrix 3.2.1. Randbedingungen des Systems 3.2.2. Verwendung einer generellen Diagonalperturbation 3.3. Iterative Lösungsverfahren 3.4. Faktorisierungsverfahren 3.4.1. Cholesky-Faktorisierung 3.4.2. LU-Faktorisierung 3.4.3. Fillin-Reduktion durch Permutation 3.4.4. Fazit 3.5. Direkte Lösungsverfahren 3.6. Verwendung externer Gleichungssystem-Löser 3.7. Zusammenfassung 4. Implementierung 4.1. Aufbau von MORPACK 4.2. Anforderungen an Reduktions-Funktionen 4.3. Eigenschaften und Optionen der KSM-Funktionen 4.3.1. Arnoldi-Funktion erster Ordnung 4.3.2. Rationale SOAR-Funktionen 4.4. Korrelationskriterien 4.4.1. Eigenfrequenzbezogen 4.4.2. Eigenvektorbezogen 4.4.3. Übertragungsfunktionsbezogen 4.5. Lösungsfunktionen linearer Gleichungssysteme 4.5.1. Anforderungen und Aufbau 4.5.2. Verwendung der Gleichungssystem-Löser 4.5.3. Hinweise zur Implementierung von Gleichungssystem-Lösern 5. Anwendung 5.1. Versuchsmodelle 5.1.1. Testmodelle kleiner Dimension 5.1.2. Getriebegehäuse 5.1.3. Treibsatzwelle 5.2. Validierung der Reduktionsmethoden an kleinem Modell 5.2.1. Modifizierte Arnoldi-Funktion erster Ordnung 5.2.2. Rationale SOAR-Funktionen 5.2.3. Zusammenfassung 5.3. Anwendung der KSM auf große Modelle 5.3.1. Getriebegehäuse 5.3.2. Treibsatzwelle 5.4. Auswertung 6. Zusammenfassung und Ausblick 6.1. Zusammenfassung 6.2. Ausblick / FEM-MKS-coupling requires model order reduction methods to simulate the frequency response of mechanical structures using a smaller reduced representation of the original system. Most of the rational Krylov-subspace methods are based on Arnoldi-algorithms. They allow to represent the frequency response in freely selectable, wide frequency ranges. Subject of this thesis is the implementation of an error-controlled model order reduction based on Krylov-subspace methods and the application to a mechanical model. Based on the MORPACK software, a first-order-Arnoldi function is extended by an interpolative start vector, the elimination of rigid body motion and a reorthogonalization. Containing these functions, a rational, interpolative Second Order Arnoldi (SOAR) method is designed that works well compared to a rational Block-SOAR-method. Interpolative equal weighting is used. The first-order-Arnoldi method requires less computational effort compared to the rational, interpolative SOAR that is able to compute a smaller reduction size for same frequency range of interest. The methods are applied to the models of a frame, a gear case and a drive shaft. Error-control is realized by eigenfrequency-based H2-integration-limit and relative H2-error based on the frequency response function. For solving linear systems of equations in Matlab, solver functions based on permutation and factorization are implemented.:1. Einleitung 1.1. Motivation 1.2. Einordnung 1.3. Aufbau der Arbeit 2. Theorie 2.1. Simulationsmethoden 2.1.1. Finite Elemente Methode 2.1.2. Mehrkörpersimulation 2.1.3. Kopplung der Simulationsmethoden 2.2. Zustandsraumdarstellung und Reduktion 2.3. Krylov Unterraum Methoden 2.4. Arnoldi-Algorithmen erster Ordnung 2.5. Arnoldi-Algorithmen zweiter Ordnung 2.6. Korrelationskriterien 2.6.1. Eigenfrequenzbezogene Kriterien 2.6.2. Eigenvektorbezogene Kriterien 2.6.3. Übertragungsfunktionsbezogene Kriterien 2.6.4. Fehlerbewertung 2.6.5. Anwendung auf Systeme sehr großer Dimension 3. Numerik linearer Gleichungssysteme 3.1. Grundlagen 3.2. Singularität der Koeffizientenmatrix 3.2.1. Randbedingungen des Systems 3.2.2. Verwendung einer generellen Diagonalperturbation 3.3. Iterative Lösungsverfahren 3.4. Faktorisierungsverfahren 3.4.1. Cholesky-Faktorisierung 3.4.2. LU-Faktorisierung 3.4.3. Fillin-Reduktion durch Permutation 3.4.4. Fazit 3.5. Direkte Lösungsverfahren 3.6. Verwendung externer Gleichungssystem-Löser 3.7. Zusammenfassung 4. Implementierung 4.1. Aufbau von MORPACK 4.2. Anforderungen an Reduktions-Funktionen 4.3. Eigenschaften und Optionen der KSM-Funktionen 4.3.1. Arnoldi-Funktion erster Ordnung 4.3.2. Rationale SOAR-Funktionen 4.4. Korrelationskriterien 4.4.1. Eigenfrequenzbezogen 4.4.2. Eigenvektorbezogen 4.4.3. Übertragungsfunktionsbezogen 4.5. Lösungsfunktionen linearer Gleichungssysteme 4.5.1. Anforderungen und Aufbau 4.5.2. Verwendung der Gleichungssystem-Löser 4.5.3. Hinweise zur Implementierung von Gleichungssystem-Lösern 5. Anwendung 5.1. Versuchsmodelle 5.1.1. Testmodelle kleiner Dimension 5.1.2. Getriebegehäuse 5.1.3. Treibsatzwelle 5.2. Validierung der Reduktionsmethoden an kleinem Modell 5.2.1. Modifizierte Arnoldi-Funktion erster Ordnung 5.2.2. Rationale SOAR-Funktionen 5.2.3. Zusammenfassung 5.3. Anwendung der KSM auf große Modelle 5.3.1. Getriebegehäuse 5.3.2. Treibsatzwelle 5.4. Auswertung 6. Zusammenfassung und Ausblick 6.1. Zusammenfassung 6.2. Ausblick

Sound criteria for partial order reduction for probabilistic concurrent systems have been presented in the literature. Their realization relies on a depth-first search-based approach for generating the reduced model. The drawback of this dynamic approach is that it can hardly be combined with other techniques to tackle the state explosion problem, e.g., symbolic probabilistic model checking with multi-terminal variants of binary decision diagrams. Following the approach presented by Kurshan et al. for non-probabilistic systems, we study partial order reduction techniques for probabilistic concurrent systems that can be realized by a static analysis. The idea is to inject the reduction criteria into the control flow graphs of the processes of the system to be analyzed. We provide the theoretical foundations of static partial order reduction for probabilistic concurrent systems and present algorithms to realize them. Finally, we report on some experimental results.

With the advances in the telecommunications industry, there is a need to solve the in-band full-duplex (IBFD) problem for antenna systems. One premise for solving the IBFD problem is to have strong isolation between transmitter and receiver antennas in an antenna system. To increase isolation, antenna engineers are dependent on simulation software to calculate the isolation between the antennas, i.e., the mutual coupling. Full-wave simulations that accurately calculate the mutual coupling between antennas are timeconsuming, and there is a need to reduce the required time. In this thesis, we investigate how implemented data-driven interpolation methods can be used to reduce the simulation times when applied to frequency domain solvers. Here, we benchmark the four different interpolation methods vector fitting, the Loewner framework, Cauchy interpolation, and a modified version of Nevanlinna-Pick interpolation. These four interpolation methods are benchmarked on seven different antenna frequency responses, to investigate their performance in terms of how many interpolation points they require to reach a certain root mean squared error (RMSE) tolerance. We also benchmark different frequency sampling algorithms together with the interpolation methods. Here, we have predetermined frequency sampling algorithms such as linear frequency sampling distribution, and Chebyshevbased frequency sampling distributions. We also benchmark two kinds of adaptive frequency sampling algorithms. The first type is compatible with all of the four interpolation methods, and it selects the next frequency sample by analyzing the dynamics of the previously generated interpolant. The second adaptive frequency sampling algorithm is solely for the modified NevanlinnaPick interpolation method, and it is based on the free parameter in NevanlinnaPick interpolation. From the benchmark results, two interpolation methods successfully decrease the RMSE as a function of the number of interpolation points used, namely, vector fitting and the Loewner framework. Here, the Loewner framework performs slightly better than vector fitting. The benchmark results also show that vector fitting is less dependent on which frequency sampling algorithm is used, while the Loewner framework is more dependent on the frequency sampling algorithm. For the Loewner framework, Chebyshev-based frequency sampling distributions proved to yield the best performance. / Med de snabba utvecklingarna i telekomindustrin så har det uppstått ett behov av att lösa det så kallad i-band full-duplex (IBFD) problemet. En premiss för att lösa IBFD-problemet är att framgångsrikt isolera transmissionsantennen från mottagarantennen inom ett antennsystem. För att öka isolationen mellan antennerna måste antenningenjörer använda sig av simulationsmjukvara för att beräkna isoleringen (den ömsesidiga kopplingen mellan antennerna). Full-wave-simuleringar som noggrant beräknar den ömsesidga kopplingen är tidskrävande. Det finns därför ett behov av att minska simulationstiderna. I denna avhandling kommer vi att undersöka hur våra implementerade och datadrivna interpoleringsmetoder kan vara till hjälp för att minska de tidskrävande simuleringstiderna, när de används på frekvensdomänslösare. Här prestandajämför vi de fyra interpoleringsmetoderna vector fitting, Loewner ramverket, Cauchy interpolering, och modifierad Nevanlinna-Pick interpolering. Dessa fyra interpoleringsmetoder är prestandajämförda på sju olika antennsystemsvar, med avseende på hur många interpoleringspunkter de behöver för att nå en viss root mean squared error (RMSE)-tolerans. Vi prestandajämför också olika frekvenssamplingsalgoritmer tillsammas med interpoleringsmetoderna. Här använder vi oss av förbestämda frekvenssamplingsdistributioner så som linjär samplingsdistribution och Chebyshevbaserade samplingsdistributioner. Vi använder oss också av två olika sorters adaptiv frekvenssamplingsalgoritmer. Den första sortens adaptiv frekvenssamplingsalgoritm är kompatibel med alla de fyra interpoleringsmetoderna, och den väljer nästa frekvenspunkt genom att analysera den föregående interpolantens dynamik. Den andra adaptiva frekvenssamplingsalgoritmen är enbart till den modifierade Nevanlinna-Pick interpoleringsalgoritmen, och den baserar sitt val av nästa frekvenspunkt genom att använda sig av den fria parametern i Nevanlinna-Pick interpolering. Från resultaten av prestandajämförelsen ser vi att två interpoleringsmetoder framgångsrikt lyckas minska medelvärdetsfelet som en funktion av antalet interpoleringspunkter som används. Dessa två metoder är vector fitting och Loewner ramverket. Här så presterar Loewner ramverket aningen bättre än vad vector fitting gör. Prestandajämförelsen visar också att vector fitting inte är lika beroende av vilken frekvenssamplingsalgoritm som används, medan Loewner ramverket är mer beroende på vilken frekvenssamplingsalgoritm som används. För Loewner ramverket så visade det sig att Chebyshev-baserade frekvenssamplingsalgoritmer presterade bättre.

A common architecture for electric vehicles is to have two electric machines one each on the front and rear axle. Despite the redundancy, this configuration ensures performance. Being energy efficient is equally important for electric vehicles to deliver a sufficiently high range. Hence, operating a single machine at low to medium torque requirement is desirable. A clutch can be implemented on the front axle and its engagement dynamically controlled to reduce the magnetic drag losses in the front machine. With clutch disengaged, the entire torque will be delivered by the rear machine causing it to heat up quickly. As electric machine and inverter losses are also temperature dependent, this work attempts to derive an optimal torque split strategy between the two machines considering thermal effects. An upper-temperature limit for both electric machine and inverter is imposed for component protection. Thermal models for the electric machine, inverter and coolant circuit are simplified using system identification and model order reduction approach. Dynamic optimal torque split is realized by minimizing the energy loss over the entire drive cycle. Dynamic programming is used to investigate the benefits of including thermal losses and to generate a benchmark solution for optimal torque split strategy. Further, two online controllers are developed, one based on non-linear model predictive control and the other being a static controller with added heuristic rules to prevent temperatures of critical components to exceed the limits. A high-fidelity plant model was developed using VSIM as master and GT-Suite thermal model as slave to compare the performance of these controllers. The results show that it is possible to obtain decent thermal performance of electric motor and inverter with one node lumped parameter thermal model and a five-node lumped parameter model for the coolant circuit. Including thermal dynamics in the controller can constraint the temperature within the limits and give an optimal torque split. The benefit of adding temperature-dependent thermal maps is found to be limited to certain operating regions. The static controller with torque split based on instantaneous power loss also performed well for the given configuration. The major contribution to energy saving was obtained by dynamic disengagement of clutch in the form of reduced magnetic drag losses. / En vanlig arkitektur för elfordon är att ha två elmaskiner en vardera på fram- och bakaxeln. Trots redundansen säkerställer denna konfiguration prestanda. Att vara energieffektiv är lika viktigt för att elfordon ska leverera en tillräckligt hög räckvidd. Det är därför önskvärt att driva en enda maskin med lågt till medelhögt vridmoment. En koppling kan implementeras på framaxeln och dess ingrepp kan styras dynamiskt för att minska de magnetiska motståndsförlusterna i den främre maskinen. Med kopplingen urkopplad kommer hela vridmomentet att levereras av den bakre maskinen vilket gör att den snabbt värms upp. Eftersom förluster av elektriska maskiner och växelriktare också är temperaturberoende, försöker detta arbete härleda en optimal vridmomentsdelningsstrategi mellan de två maskinerna med tanke på termiska effekter. En övre temperaturgräns för både elektrisk maskin och växelriktare är införd för komponentskydd. Termiska modeller för den elektriska maskinen, växelriktaren och kylvätskekretsen förenklas med hjälp av systemidentifiering och modellbeställningsreduktion. Dynamisk optimal vridmomentdelning realiseras genom att minimera energiförlusten under hela körcykeln. Dynamisk programmering används för att undersöka fördelarna med att inkludera termiska förluster och för att generera en benchmarklösning för optimal vridmomentsdelningsstrategi. Vidare utvecklas två online-styrenheter, en baserad på icke-linjär modell för prediktiv styrning och den andra är en statisk styrenhet med tillagda heuristiska regler för att förhindra att temperaturer på kritiska komponenter överskrider gränserna. En högfientlig anläggningsmodell utvecklades med VSIM som master och GT-Suite termisk modell som slav för att jämföra prestandan hos dessa styrenheter. Resultaten visar att det är möjligt att erhålla hyfsad termisk prestanda för elmotor och växelriktare med en termisk modell med en nodklumpad parameter och en femnodsmodell med klumpparametrar för kylvätskekretsen. Att inkludera termisk dynamik i regulatorn kan begränsa temperaturen inom gränserna och ge en optimal vridmomentfördelning. Fördelen med att lägga till temperaturberoende termiska kartor har visat sig vara begränsad till vissa driftsområden. Den statiska styrenheten med vridmomentdelning baserad på momentan effektförlust fungerade också bra för den givna konfigurationen. Det största bidraget till energibesparingen erhölls genom dynamisk urkoppling av kopplingen i form av minskade magnetiska motståndsförluster.