Algorithmic System Design under Consideration of Dynamic Processes

Vom Fachbereich Maschinenbau an der Technischen Universit¨atDarmstadt zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften (Dr.-Ing.)

genehmigte Dissertation

vorgelegt von

Dipl.-Phys. Lena Charlotte Altherr

aus Karlsruhe

Berichterstatter: Prof. Dr.-Ing. Peter F. Pelz Mitberichterstatter: Prof. Dr. rer. nat. Ulf Lorenz Tag der Einreichung: 19.04.2016 Tag der m¨undlichen Pr¨ufung: 24.05.2016

Darmstadt 2016

D 17

Vorwort des Herausgebers

Kontext Die Produkt- und Systementwicklung hat die Aufgabe technische Systeme so zu gestalten, dass eine gewünschte Systemfunktion erfüllt wird. Mögliche System- funktionen sind z.B. Schwingungen zu dämpfen, Wasser in einem Siedlungsgebiet zu verteilen oder die Kühlung eines Rechenzentrums. Wir Ingenieure reduzieren dabei die Komplexität eines Systems, indem wir dieses gedanklich in überschaubare Baugruppen oder Komponenten zerlegen und diese für sich in Robustheit und Effizienz verbessern. In der Kriegsführung wurde dieses Prinzip bereits 500 v. Chr. als „Teile und herrsche Prinzip“ durch Meister Sun in seinem Buch „Die Kunst der Kriegsführung“ beschrieben. Das Denken in Schnitten ist wesentlich für das Verständnis von Systemen: „Das wichtigste Werkzeug des Ingenieurs ist die Schere“. Das Zusammenwirken der Komponenten führt anschließend zu der gewünschten Systemfunktion. Während die Funktion eines technischen Systems i.d.R. nicht verhan- delbar ist, ist jedoch verhandelbar mit welchem Aufwand diese erfüllt wird und mit welcher Verfügbarkeit sie gewährleistet wird. Aufwand und Verfügbarkeit sind dabei gegensätzlich. Der Aufwand bemisst z.B. die Emission von Kohlendioxid, den Energieverbrauch, den Materialverbrauch, … die „total cost of ownership“. Die Verfügbarkeit bemisst die Ausfallzeiten, Lebensdauer oder Laufleistung. Die Gesell- schaft stellt sich zunehmend die Frage, wie eine Funktion bei minimalem Aufwand und maximaler Verfügbarkeit realisiert werden kann.

Die Methode Die Antwort auf die oben genannte Frage besteht darin, nicht nur nach dem Prinzip „Teile und herrsche“ einzelne Komponenten des Gesamtsystems isoliert zu optimie- ren, sondern deren Zusammenspiel zu berücksichtigen. Das Spielfeld für den Entwurf der Kühlung eines Rechenzentrums besteht z.B. aus Rohrleitungen, Pumpen unter- schiedlicher Größe, Ventilen und Wärmetauschern. Für ein optimales Gesamtsystem ist es Aufgabe, die Komponenten aus dem Spielfeld so anzuordnen, zu verschalten und zu betreiben, dass die Funktion mit minimalem (Energie-)Aufwand und maximaler Verfügbarkeit erfüllt werden kann. Für einfache Systeme gelingt dem Planer die optimale Systemsynthese, für größere Systeme ist dies ohne algorithmische Hilfe unmöglich. Der Grund liegt in der für den Menschen unüberschaubaren Zahl möglicher Entschei- dungen. Die hinter den Entscheidungen steckenden Fragen sind immer diskret, d.h. der Art „Komponente A oder B?“. Bei einer Verschaltung von 6 Komponenten eines Pumpensystems sind bereits über 5000 Systemvarianten möglich. Motivation des Technical Operations Research (TOR) ist es, Algorithmen als Entscheidungsunter- stützung zu nutzen, um optimale Systeme zu planen und zu betreiben. Der TOR-Ansatz wurde für quasistationäre und quasistatische Probleme in den vergangenen vier Jahren an der TU Darmstadt entwickelt und erfolgreich zur Struk- turierung von Fluidfördereinrichtungen (Booster-Station) angewendet. Der TOR-

I Ansatz wurde bisher nicht auf zeitvariante Probleme angewendet. Zeitvariante Prob- leme treten jedoch immer auf, wenn die Verfügbarkeit von Systemen im algorith- mischen Entwurfsprozess antizipiert werden soll. Dabei geht es um die Antizipation von Systemdegradation durch mechanische und/oder chemische Alterung.

Die Forschungsfrage Frau Altherr erweitert den TOR-Ansatz auf zeitvariante Probleme und zeigt dessen Anwendung am Beispiel eines hydraulischen Systems. Hierbei ist neu, dass die durch den späteren Betrieb zu erwartende Systemdegradation eines jeden möglichen Systemvorschlags bereits in der Entwicklungsphase antizipiert werden kann. Dadurch können Betriebs- und Wartungskosten vorausgesagt und minimiert werden und durch eine optimale Betriebs- und Wartungsstrategie die Verfügbarkeit des Systems garantiert werden. Insgesamt ergibt sich ein mehrstufiges Optimierungsproblem, welches sich nicht direkt mit Standardsoftware lösen lässt. Während sich die erste Stufe des Optimierungsmodells durch eine sehr hohe kombinatorische Lösungsvielfalt bei schon wenigen zur Verfügung stehenden Komponenten auszeichnet, muss in der zweiten Stufe komplexe nichtlineare Physik berücksichtigt werden. Frau Altherr stellt sich in der Arbeit zwei Aufgaben, eine technologische und eine methodische. Methodisch lotet sie diskrete Optimierungsmethoden zur Strukturie- rung von zeitvarianten Systemen aus, die durch Zeitglieder bis maximal erster Ord- nung beschrieben werden. Damit lässt sich die Systemdegradation durch mechani- schen und chemischen Verschleiß, sowie das zeitvariante Verhalten von Speicher- elementen gleichermaßen behandeln. Technologisch wendet Frau Altherr gemischt-ganzzahlige lineare Programmierung an, um die Abtriebsseite eines hydrostatischen Getriebes algorithmisch zu strukturieren. Wesentliche Fragen sind hierbei die physikalische Modellierung, die Darstellung des Optimierungsproblems als mathematisches Programm, und dessen algorithmische Behandlung zur Lösungsfindung. Frau Altherr setzt hierzu einerseits auf Heuristiken zum schnelleren Auffinden sinnvoller Topologien, andererseits auf mathematische Dekomposition zur Bewertung des dynamischen Verschleiß- und Wartungsverlaufs.

Ergebnisse Mit den von Frau Altherr vorgestellten Modellen und Algorithmen ist es möglich, Systemstrukturen zu finden, die eine 16-fach höhere Lebensdauer als ein konven- tioneller Systementwurf aufweisen. Wird die erwartete Lebensdauer maximiert, entstehen Systeme mit aktiver Redundanz. Die redundanten Komponenten werden genutzt, um die Last intelligent zu verteilen und damit das Gesamtsystem langlebiger zu machen, bei minimal möglichem Mehrinvest. Nachteil der entstehenden Systeme kann ihre höhere Komplexität sein, wodurch der Aufwand steigt. Frau Altherr zeigt, wie zusätzlich eine auf realistische Bedingungen zugeschnittene Abwägung zwischen Aufwand und Verfügbarkeit bereitgestellt werden kann. Dazu erweitert sie das vorgestellte Grundmodell um die Betrachtung von Wartungsarbeiten.

II Verschlissene Komponenten können zu bestimmten Wartungszeitpunkten aus- getauscht werden, so dass das System weiter seine Funktion erfüllen kann. Die Wartungsintervalle führen zu mehrstufigen mathematischen Programmen, deren algorithmische Behandlung neue Herausforderungen schafft. Frau Altherr entwickelt eine Primalheuristik, die bei der gegebenen kombinatorischen Vielfalt verschiedenster Systemaufbauten dennoch schnell zu sinnvollen Topologie- vorschlägen gelangt. Grundlage ist die Idee, sich zunächst auf serien-parallele Verschaltungen der Komponenten zu konzentrieren, da diese mittels Konstruktions- vorschriften systematisch aufgestellt werden können. Das Gesamtnetzwerk des hydro- statischen Getriebes wird in der entwickelten Heuristik anschließend als Kombination vierer serien-paralleler Teilnetzwerke aufgebaut. Frau Altherr implementiert eine Nachbarschaftsssuche, welche aus-gehend von einer validen Lösung den Raum der diskreten Systemstrukturen systematisch durchsucht. Diese Lokale Suche wird von Frau Altherr in die Metaheuristik Simulated Annealing eingebettet, sodass einem Verharren des Algorithmus in lokalen Optima entgegengewirkt werden kann. Zum Auffinden einer optimalen Betriebs- und Wartungsstrategie und von dualen Schran- ken implementiert Frau Altherr Dynamische Programmierung und ein Langrange- Dekompositions-Verfahren.

Innovation Durch die Arbeit wird eine neue Dimension der Strukturoptimierung aufgezeigt. Man wird in Zukunft bei der Gestaltung von Systemstrukturen neben der reinen Funktions- erfüllung deutlich besseren Zugang zu Sekundärkriterien wie Lebensdauer und Verfügbarkeitsgarantien erhalten. Die vorliegende Schrift beinhaltete folgende Inhalte und Teile . Darstellung des TOR-Gesamtgedankens über die gesamte Schrift hinweg . Erstellung eines physikalisch hinterlegten technischen Optimierungsmodells, welches neben der Fluidmechanik Verschleiß kodiert und Wartungs- optimierung auf formaler Ebene ermöglicht . Aufbau eines Gesamtkonzepts von hochformaler mathematischer Modellierung über heuristische Lösungsfindung plus Optimalitäts- abschätzungen über duale Schranken in einem bislang eher schwer zugänglichen Bereich der technischen Systeme Die Optimierung technischer Systeme an der Schnittstelle von Mathematik, Informatik und Ingenieurwesen wird sowohl gründlich als auch anschaulich und nachvollziehbar dargestellt. Ich bin der Überzeugung, dass die vorliegende Arbeit die algorithmische Herangehensweise zum Lösen von Nachhaltigkeitsproblemen verschiedener Art nachhaltig beeinflussen wird.

Peter Pelz, Darmstadt, am 22.08.2016

III

Vorwort

Ich m¨ochte allen danken, die zum Gelingen dieser Arbeit beigetragen haben. Mein besonderer Dank hierbei gilt: Prof. Dr.-Ing. Peter F. Pelz, der mir f¨urdie Durchf¨uhrungdieser Arbeit vollste Unterst¨utzungzukommen ließ und mir durch seine interessante Aufgabenstellung interdisziplin¨aresForschen an der Schnittstelle zwischen Ingenieurwissenschaften und Mathematik erm¨oglicht hat. Prof. Dr. Ulf Lorenz f¨urdie Zweitkorrektur und die sehr gute fachliche Betreuung des mathematischen Aspekts dieser Arbeit. Durch inhaltliche Diskussionen mit ihm hat meine Arbeit an thematischer Vielfalt gewonnen. Dem Leiter der Gruppe Technische Optimierung, Thorsten Ederer, f¨urdie Durch- sicht dieser Arbeit und die große Unterst¨utzungbei fachlichen und praktischen Fragen. Dem gesamten Team der Gruppe Technische Optimierung: Philipp P¨ottgen,Mar- lene Utz und Jan Wolf, die mir in Gespr¨achen stets hilfreiche Anregungen geben konnten. Allen Kollegen und Studierenden am Institut f¨urFluidsystemtechnik, die zum Entstehen dieser Arbeit beigetragen haben. Insbesondere danke ich Angela Verg´e, die durch ihr gemeinsam mit Prof. Pelz entwickeltes Modell zum Verschleiß von Ventilen durch str¨omungsbedingten Materialabtrag einen wichtigen Beitrag zu dieser Arbeit geleistet hat. Ganz besonderer Dank gilt meinen Eltern, meiner Oma und meinem Partner Patrick. Ich m¨ochte ihnen an dieser Stelle von ganzem Herzen f¨urihre bedin- gungslose Unterst¨utzungdanken.

Lena Altherr Eppelheim, im April 2016

V

Abstract

It is common practice to optimize technical components in terms of their charac- teristics, such as energy efficiency or durability. However, even highly optimized components can lead to inefficient or unstable system structures if their interaction is not taken into account. Considering systems instead of components opens up further potential for optimization. However, the increased potential is accompanied by an increased level of complexity. This work was written as part of the Collaborative Research Centre 805, whose aim is to control uncertainty in load-carrying structures. In this context, the question of finding an optimal system structure has been investigated for a truss topology demonstrator. This dissertation examines the issue of structural uncertainty for a hydraulic system. The structure of a hydrostatic transmission system whose components are sub- ject to erosive wear is optimized via Mixed-Integer Linear Programming. Espe- cially system synthesis problems in which dynamic processes like wear have to be modeled can often not be solved by standard discrete optimization techniques. This work exploits their specific problem structure to derive primal solutions and dual bounds. The optimization problem is based on the graph representation of the system syn- thesis problem. In case of a series-parallel network, all possible system graphs can be represented by trees without internal nodes of degree 2 (series-reduced trees). This work shows how the approach can be applied to a non-series-parallel net- work by using problem-specific knowledge to divide the graph into series-parallel subgraphs. In this way, the tree representation can still be exploited to elegantly traverse the solution space of discrete structure decisions using a local search. Based on a combination of simulated annealing and dynamic programming, a pri- mal heuristic is developed that allows to find good solutions to the optimization problem when commercial solvers cannot find any feasible solution in reasonable time. By applying Bellman’s principle of optimality, an optimal maintenance plan in case of wear induced component failures is derived. Dual decomposition is imple- mented to decouple the sequence of operation intervals in between service points, and to derive replacement costs by dual bounds. By means of the models and algorithms developed, wear-related maintenance costs and downtime of a possible system layout can be anticipated in the early phase of structure determination. This thesis thereby lays a foundation for the control of structural uncertainty in real systems.

VII

Zusammenfassung

Nach Stand von Wissenschaft und Technik werden Komponenten hinsichtlich ihrer Eigenschaften, wie Lebensdauer oder Energieeffizienz optimiert. Allerdings k¨onnen selbst hervorragende Komponenten zu ineffizienten oder instabilen Systemen f¨uhren, wenn ihr Zusammenspiel nur unzureichend ber¨ucksichtigt wird. Eine Systembetra- chtung schafft ein gr¨oßeresOptimierungspotential — dem erh¨ohten Potential steht jedoch auch ein erh¨ohter Komplexit¨atsgradgegen¨uber. Durch die Kombinatio- nen unterschiedlicher Komponenten zu Systemen existiert eine Vielzahl m¨oglicher Strukturen, wodurch sich eine kombinatorische Explosion des L¨osungsraumsergibt. Die vorliegende Arbeit ist im Rahmen des Sonderforschungsbereichs 805 entstanden, dessen Ziel die Beherrschung von Unsicherheit in lasttragenden Systemen ist. Die Frage nach der optimalen Struktur eines Systems wurde in diesem Zusammenhang bereits f¨urden Fall eines Stabwerk-Demonstrators untersucht. Diese Arbeit zeigt anhand eines realen Systems aus dem Bereich der Hydraulik, wie Strukturunsicher- heit durch Antizipation in der Entwicklungsphase beherrscht werden kann. Ein hydrostatisches Getriebe, dessen Komponenten Verschleiß ausgesetzt sind, wird mittels gemischt-ganzzahliger linearer Programmierung optimal strukturiert. Das zugeh¨orige Optimierungsmodell basiert auf der Graph-Darstellung des Struk- turfindungsproblems. Diese ist im Falle eines serien-parallelen Netzwerkes durch B¨aumeohne innere Knoten des Grads zwei darstellbar. Die Arbeit zeigt f¨urden all- gemeineren Fall eines nicht serien-parallelen Netzwerkes, wie sich diese Darstellung durch Unterteilung des Graphen in serien-parallele Teilgraphen dennoch nutzen l¨asst,um in einer Lokalen Suche im L¨osungsraumder diskreten Strukturentschei- dungen valide Topologievorschl¨agezu erhalten. Gerade Strukturfindungsprobleme, in denen zus¨atzlich dynamische Prozesse wie Verschleiß abgebildet werden sollen, k¨onnen mit Standardsolvern oft nicht in ange- messener Zeit global optimal gel¨ostwerden. Die vorliegende Arbeit nutzt die spezielle Struktur solcher Probleme, um mittels Dekompositionsverfahren primale L¨osungenund duale Schranken abzuleiten. Durch eine Primalheuristik, die auf der Metaheuristik Simulated Annealing basiert, k¨onnenauch dann gute L¨osungenf¨ur das Optimierungsproblem gefunden werden, wenn kommerzielle Solver ¨uberhaupt keine zul¨assige L¨osungin annehmbarer Zeit finden. M¨ogliche Systemstrukturen werden dabei durch Dynamische Optimierung bewertet. Unter Ausnutzung des Bellman’schen Optimalit¨atsprinzipswird durch R¨uckw¨artsrekursion eine optimale Wartungsstrategie bei Verschleißprozessen berechnet. Uber¨ duale Dekomposition werden zudem zur Absch¨atzungder Ersatzteilkosten duale Schranken abgeleitet. Mittels der vorgestellten Modelle und den entwickelten Algorithmen lassen sich verschleißbedingte Wartungskosten und Ausfallzeiten einer Systemvariante bereits in der fr¨uhenPhase der Strukturfindung antizipieren. Diese Dissertation legt damit den Grundstein zur Beherrschung von Strukturunsicherheit in realen Systemen.

IX

Contents

List of Figures XIII

Symbol Directory XV

1 Introduction 1 1.1 The TOR Ansatz ...... 1

2 Mathematical Basics 5 2.1 Mixed-Integer Programming ...... 5 2.2 Primal, Dual, and the Optimality Gap ...... 6 2.3 Solving to Global Optimality ...... 7

3 Optimization of Technical Systems 9 3.1 Optimization of Fluid Systems ...... 9 3.2 Special Problem Structure of System Synthesis Problems ...... 11

4 Wear Processes and Algorithmic System Design 13 4.1 Mathematical Optimization Model for the Hydrostatic Transmission System ...... 13 4.2 Extension of the Model ...... 17

5 Topology Optimization 21 5.1 Superstructures of Possible Layouts ...... 21 5.2 Number of Possible Topology Options ...... 21 5.3 Topology Options for the Hydrostatic Transmission System . . . . . 24 5.3.1 Series-Parallel Networks ...... 26 5.4 Local Search for System Topologies ...... 29 5.4.1 Neighborhood Function ...... 31 5.5 Simulated Annealing ...... 36 5.6 Implementation and Benchmarks ...... 38

6 Dynamic Programming 41 6.1 Fundamentals ...... 41 6.2 Application to Dynamic Decision Problems ...... 42 6.3 Algorithmic Operations ...... 45 6.3.1 Benchmarks ...... 48

XI Contents

7 Dual Bounds via Lagrangean Relaxation 51 7.1 Basics of Lagrangean Relaxation ...... 51 7.1.1 Subgradient Method ...... 53 7.2 Lagrangean Decomposition ...... 54 7.3 Wear-Induced Component Replacement ...... 56 7.3.1 Dual Decomposition Approach ...... 56 7.3.2 Verification For Small Test Cases ...... 58 7.3.3 Benchmarks ...... 60

8 Conclusion and Outlook 65

Bibliography 67

XII List of Figures

1.1 TOR pyramid illustrating the seven steps of the TOR ansatz. . . .3

2.1 Convex and non-convex sets...... 7

3.1 Problem Structure for a system synthesis problem...... 10 3.2 Problem Structure for a system synthesis problem considering dy- namic processes ...... 11

4.1 Details on the hydrostatic transmission system...... 15 4.2 Optimal topology of the hydrostatic transmission system for a max- imum lifespan ...... 17 4.3 Optimal system structures for the hydrostatic transmission system. 20

5.1 Exemplary system topologies for a hydrostatic transmission system. 24 5.2 Runtimes of the commercial solver IBM ILOG CPLEX for the prob- lem of finding an optimal system structure for the hydrostatic trans- mission system...... 25 5.3 Graph of the hydrostatic transmission system considering only open valves in each load case...... 27 5.4 Graph representing superstructure of all considered system layouts. 28 5.5 Series-parallel networks of order 1 to 4...... 29 5.6 Exemplary executions of the swap neighborhood function...... 32 5.7 Exemplary executions of the replace neighborhood function. . . . . 33 5.8 Exemplary executions of the switch neighborhood function...... 34 5.9 Exemplary executions of the add neighborhood function...... 34 5.10 Exemplary executions of the delete neighborhood function...... 35 5.11 Progress of the solution quality obtained via simulated annealing for 10 operation intervals...... 38 5.12 Results for the simulated annealing heuristic for the case of three operation intervals...... 39

6.1 Basic structure for deterministic dynamic programming...... 43 6.2 Structure of a dynamic program representing a decision problem containing 3 wear levels and 2 valves, and start of the solution pro- cedure...... 45 6.3 Second step of the solution procedure in the dynamic program. . . . 46

XIII List of Figures

6.4 The dynamic optimization yields an optimal policy from stage 0 to stage T ...... 47 6.5 Runtime of the dynamic programming algorithm for the test case of four valves...... 49 6.6 Run time of the dynamic program for 10 service intervals for different numbers of valves and different numbers of levels...... 49

7.1 Graphical interpretation of the Lagrangean relaxation...... 52 7.2 Graphical interpretation of the Lagrangean decomposition. . . . . 55 7.3 Evolution of wear, and replacement strategy for a simple test case with one valve...... 60 7.4 Evolution of the dual bound with the number of iterations for dif- ferent test cases...... 62 7.5 Evolution of the dual bound for two valves and different numbers of operation intervals...... 63

XIV Symbol Directory

Symbols are listed in the first column, and are described in the second column. The third column gives the corresponding dimension if it can be stated, with the base factors length L, mass M, and time T . Symbol Description Dimension A Coefficient matrix of a general optimization prob- lem. 2 Apiston Piston area. L

αvi,vj Binary variable indicating whether valve (vi, vj) is shut or open. b Right-hand vector of a general optimization prob- lem. bi,j Binary variable indicating whether a component (i, j) is purchased or not. c Coefficient vector of objective function. c(k) Control parameter of simulated annealing algo- rithm in iteration k. Cinvest Investment costs. Cmaint Maintenance costs.

Cvi,vj Price of valve (vi, vj). cV Volume fraction of suspended solids. d Valve diameter. L E Set of edges of a graph. Ej Energy of a state j in the Metropolis algorithm. −2 Fl Required piston force in load case l. MLT fn(sn) Contribution of stages n, n+1,...,N to the objec- tive function of a dynamic optimization problem if system starts in state sn at stage n. G Graph. G(V,E) Graph with vertices V and edges E. Gl Graph for load case l. g Subgradient of the Lagrangean function. g(k) Subgradient of the Lagrangean function in itera- tion k of the subgradient method. (i, j) Edge of a graph. k Iteration number.

XV Symbol Directory

kB Boltzmann constant. −1 l,vi,vj Reciprocal time constant for wear of valve (vi, vj) T K depending on operating parameters of load case l. l Load case index. L Set of load case indices. L(λ) Lagrangean function. λ Vector of Lagrange multipliers. λ(0) Initial vector of Lagrange multipliers in the first iteration of the subgradient method. λ(k) Vector of Lagrange multipliers in iteration k of the subgradient method. µ(k) Step size in iteration k of the subgradient method. n Index for stages in a dynamic optimization prob- lem. N Number of stages in a dynamic optimization prob- lem. p Pressure. L−1MT −2 −1 −2 pmax Upper bound for pressure. L MT ∆p Pressure decrease. L−1MT −2 π Circle constant. Π Combinatorial optimization problem. Powerset. QP Volume flow. L3T −1 3 −1 Qmax Upper bound for volume flow. L T

rs,vi,vj Binary variable indicating if valve (vi, vj) is re- placed at maintenance point s. R The set of real numbers. ρ Oil density. ML−3 s Operation interval index. sbest Current best solution found by Simulated Anneal- ing. sinitial Initial solution the Simulated Annealing algorithm is provided with. sn Current state for stage n in a dynamic optimiza- tion problem. Sn Set of reachable states for stage n in a dynamic optimization problem. s˜, t˜ Terminals of a graph. t Time step. T Index for last operation interval. T1,...,T4 The four series-reduced trees representing the valve system. tl Duration of load case l. T

XVI Symbol Directory

trn(sn, xn) Transformation function of a dynamic optimiza- tion problem which maps the current state and decision to a new state sn+1. umax Maximum valve lift. L u Valve lift. L V Set of vertices of a graph. Set of edges (v , v ) representing valves in a graph Vl i j Gl corresponding to load case l. prop Set of edges representing proportional valves. V Set of edges representing proportional valves of Vprop,l graph Gl. −1 v˜l Required piston velocity in load case l. LT (vi, vj) Edge representing a valve. wmax Upper bound for the accumulated wear. L ws,vi,vj Wear of the control edge of a proportional valve L (vi, vj) that is accumulated until the end of oper- ation interval s. begin ws,vi,vj Accumulated wear of the control edge of a propor- L tional valve (vi, vj) present at the start of operation interval s. end ws,vi,vj Accumulated wear of the control edge of a propor- L tional valve (vi, vj) present at the end of operation interval s.

∆wvi,vj Wear of the control edge of a proportional valve L (vi, vj). x Vector of all decision variables of a general opti- mization problem. x∗ Optimal solution to a general optimization prob- lem. ys,vi,vj Auxiliary variable for each coupling constraint L that represents the common value of the coupled variables for valve (vi, vj) in operation interval s. yˆs,vi,vj Average of the variables decoupled by Lagrangean L decomposition for valve (vi, vj) in operation inter- val s. z Objective value of a general optimization problem. z∗ Optimal objective value of a general optimization problem. zdual Dual bound. zLD Dual bound obtained by Lagrangean decomposi- tion. zLP Dual bound obtained by LP relaxation. zLR Dual bound obtained by Lagrangean relaxation. zprimal Primal bound.

XVII Symbol Directory

zSim.Ann. Objective value obtained by Simulated Annealing. zub General upper bound on the objective value. Z The set of integers. ζ Pressure loss coefficient.

XVIII 1 Introduction

It is common practice to optimize technical components in terms of their charac- teristics, such as energy efficiency or durability. However, even highly optimized components can lead to an inefficient or unstable system structure if their interac- tion is not taken into account. Considering systems instead of components opens up further potential for optimization. A holistic approach for the design of technical systems is pursued by the Technical Operations Research (TOR) Ansatz [PLL14] which is outlined in this chapter and which is the basis of this thesis. The TOR Ansatz has been successfully applied to various technical systems, e.g., to find the optimal design of booster stations [Alt+16], ventilation systems [Sch+], or heating systems [P¨ot+15b].Especially in the context of uncertainty, e.g., due to unexpected component failures or uncertainties in the expected load, the optimal system topology is often unclear. TOR has also been successfully applied in these cases. For a booster station, an optimization model was developed that anticipates future downtime costs due to stochastic failures [Alt+15b], while uncertainty in the expected load profile was considered in [P¨ot+15a]. In this thesis, the application of TOR to a technical system in the context of the Collaborative Research Centre 805 – Control of Uncertainty in Load-Carrying Structures in Mechanical Engineering is investigated: The task is to find the optimal topology and operation of a hydrostatic transmission system. Possible layout options consist of different switch and proportional valves and all of their possible connections. Since it has been found for hydraulic valves that a contamination of the hydraulic fluid by solid particles is the main cause of failure [Sch04], the focus in this application example lies on the dynamic process of erosive wear of the system components.

1.1 The TOR Ansatz

When determining a reasonable system structure, i.e., finding the right selection and arrangement of components, so that the desired task can be fulfilled, the designer has to make several discrete decisions: “Should I buy this component or another one?”, “Should I buy one, two or three (possibly redundant) components?”,

1 1 Introduction

“How should the components be connected?”, ... Even for what seems to be a relatively small amount of possible component options, these decisions lead to a multitude of possible system structures, also known as combinatorial explosion.

Computer support is however often only used in the form of simulations of pre- determined single system structures. Whenever discrete decisions come into play, standard gradient based optimization techniques can only be applied if one enu- merates all possible options and optimizes each one separately - an approach that is not feasible in practical system design, since the number of possible design options can quickly reach magnitudes of thousands.

In contrast, the TOR Ansatz aims at considering a superset of possible system structures simultaneously. Instead of performing parameter optimization for spe- cific, pre-defined system structures, the structure itself is already put in question. The system designer is supported by algorithms known from the field of discrete optimization, that find the optimal topology option and its optimal operation in the plurality of possible system variations.

TOR provides a systematic guideline for the optimal design of technical systems. Fig. 1.1 schematically depicts the single steps. The first step is to clarify the basic task of the system. Technical requirements are determined by the specification of a load profile that has to be satisfied. Subsequently, the system designer has to specify his objective which is always subjective. His goal may be to find a system topology, which fulfills the required function and incurs lowest possible investment costs. He could however also prefer a system structure that requires the lowest energy consumption during the expected runtime of the system. Combinations of different, possibly opposite objectives are also conceivable, e.g., when consid- ering the conflict of additional investment costs versus possibly increased system availability.

The third step is the determination of the degrees of freedom of the topology optimization. This includes answering the following questions: “Out of which kit of optional components will the optimization algorithm be free to choose?”, “Do we e.g., consider different pump types or only equal pump types but different models?”, “What design alternatives are imaginable?”, “What degree of freedom is there in linking the optional components?”, “Are only series or parallel connections possible or do we want to consider all possible connections?” In this step, the physical relationships of the system are modeled. This includes characteristic curves of all of the optional components, as well as system wide conservation laws. By concluding this step, all necessary input data for the topology optimization is set.

In a further step, a Mixed-Integer Nonlinear Optimization Problem (MINLP) is created. This MINLP is piecewisely linearized to obtain a Mixed-Integer Lin- ear Optimization Problem (MILP). It contains not only continuous variables, but also integer variables that represent the discrete structure decisions. The decision

2 Technical Operations Research - TOR 1.1 The TOR Ansatz

1. WHAT IS THE TASK?

2. WHAT IS THE OBJECTIVE?

3. WHAT ARE THE DEGREES OF FREEDOM?

4. OPTIMIZE!

5. VERIFY!

6. VALIDATE!

7. REALIZE!

20. NovemberFigure 2013 1.1 | FAA –Forschung TOR | Prof. pyramid Dr.-Ing. Peter illustrating Pelz, PD Dr. Ulf theLorenz seven| TOR steps of the2 TOR ansatz. whether a particular component is ideally be installed or not, can e.g., be inte- grated into the optimization model by a binary variable. Section 2 gives a short overview of the mathematical basics. Once, an optimal system structure has been found, the underlying optimization model has to be verified, i.e., it has to be checked whether the relevant functional relations in the system were represented correctly and whether the model was solved numerically correctly. If this is the case, validation can take place, i.e., checking to what extent the model and its detail are suitable for the description of the relevant physical behavior and dependencies. This can e.g., be done by numerical simulations where the found optimal system structure is fixed. If possible, the optimization results can also be validated by comparison with measurements on real systems. For the case of a booster station, an experimental validation in [Alt+16] showed that in the worst case the predicted energy consumption of an optimal system topology only differed by 10% from the found optimum.

3 1 Introduction

4 2 Mathematical Basics

Many real-world optimization problems consist of a mixture of discrete decisions and physical relationships that are described by continuous variables. In the case of algorithmic system design, discrete decisions arise from investment decisions and e.g., from shutting and opening shift valves during operation. Continuous variables describe physical quantities in the system like pressure or volume flow, and continuous operation parameters like the valve lift of a proportional valve or the rotational speed of a pump. Such problems can be modeled by mixed-integer linear programs which will be briefly introduced in this section. A standard reference is [Sch98]. The outline given in this chapter follows the short introduction of [F¨ug+10].

2.1 Mixed-Integer Programming

A Mixed-Integer Linear Problem (MILP) is a mathematical optimization problem consisting of a set of variables, and a set of linear constraints on these variables. Furthermore, a set of integrality constraints on some of the variables is given, spec- ifying that these variables must assume integer values. Goal of the optimization is to find realizations of the variables that satisfy both the linear and the integral- ity constraints and minimize or maximize a given linear objective function (e.g. min cT x). A maximization problem can be easily transformed into a minimiza- tion problem by multiplying the objective function by 1. All definitions and theorems can also be easily transferred. Therefore, in this− work, we will focus on minimization problems. This leads to the following formal notation for a MILP:

min cT x s.t. Ax b (2.1) ≤ x Zp Rn−p. ∈ × where A Qm×n, b Qm, c Qn with m, n, p N, p n is the input data ∈ ∈ ∈ ∈ ≤ describing a certain instance. If a vector x Zp Rn−p satisfies Ax b it is called a feasible solution for Problem 2.1. Solving∈ Problem× 2.1 to global optimality≤ means to find a feasible solution x∗ Zp Rn−p that minimizes the objective function ∈ × such that for any other feasible solutionx ˆ Zp Rn−p it holds cT x∗ cT xˆ. ∈ × ≤

5 2 Mathematical Basics 2.2 Primal, Dual, and the Optimality Gap

In some cases, solving an optimization problem to global optimality is not possible within reasonable time. In this case, one would want to conclude how far away the best solution found so far is from the global optimum. By tackling the problem from two sides, the primal and the dual, a so-called optimality gap can be derived. It allows to assess the difference between the incumbent best known solution and a value that bounds this best possible solution.

At the primal side one is interested in finding feasible solutions within a short amount of time. Common primal methods which have been used in a broad field of applications are meta-heuristics like genetic algorithms, tabu search or simu- lated annealing. While having the ability of finding good solutions quickly, it is in principle not possible to guarantee global optimality or derive the size of the optimality gap. This is where the dual side comes into play.

The main strategy of dual methods is to relax the original Problem 2.1 such that it can be solved more efficiently. Some “hard” constraints of the problem are dropped and it is solved to global optimality. Since all feasible solutions to the original Problem 2.1 are also feasible solutions of the relaxed Problem 2.2, this gives a ∗ lower bound zdual on an optimal objective value z of the original Problem.

A general strategy in mixed-integer programming is to relax the integrality condi- tion and solve the corresponding linear programming (LP) relaxation:

min cT x s.t. Ax b (2.2) ≤ x Rn. ∈ The set of feasible solutions to an LP is a polyhedron, which can be described by an intersection of a finite set of halfspaces, i.e., it is a convex subset of Rn, c.f. Fig. 2.1. A polyhedron can also be described by the Minkowski sum of a convex hull of finitely many points and a cone of finitely many rays. The convex hull conv(M) of a subset M of Rn is the smallest possible convex set that contains M, namely the intersection of all convex sets containing M.

Due to the convexity of the feasible set and the linearity of the objective function the optimal solution to Problem 2.2 – if it exists – can always be found on a corner of the polyhedron, i.e., on an extremal point of the convex hull. Problem 2.2 can be solved to global optimality by a linear programming algorithm, such as the simplex algorithm of Dantzig [Dan51]. It will either return a verified global optimal solution (which does not have to be unique), or verify that the optimization problem is unbounded (i.e., the direction of the objective function follows the direction of the cone) or infeasible (i.e., the polyhedron is an empty set).

6 2.3 Solving to Global Optimality

CONVEX SET NON-CONVEX SET

CONVEX POLYTOPE CONVEX POLYHEDRON = POLYTOPE + CONE

Figure 2.1 – Convex and non-convex sets. For every pair of points within a convex set, every point on the straight line segment that joins the pair of points is also within the region. A polytope is a bounded polyhedron. A polyhedron can be described by the sum of a polytope and a cone.

We have seen that the objective value of each dual solution zdual is a lower bound to the global optimal solution z∗ of the original problem. On the other hand, the objective value zprimal of each feasible solution of the primal problem is an upper bound to the global optimal solution:

z z∗ z (2.3) dual ≤ ≤ primal

This allows us to define the optimality gap as follows:

z z gap := primal − dual . (2.4) zdual

It measures the distance between primal and dual bound, however we cannot derive where the global optimal solution z∗ lies in respect to both bounds. Our goal is therefor to improve dual as well as primal bound until we reach an optimality gap of (almost) zero.

2.3 Solving to Global Optimality

If one can show that the optimality gap is zero, one has found a global optimal solution to the MILP 2.1. Therefore, it is necessary to improve both primal and dual solutions.

7 2 Mathematical Basics

The lower dual bound can be tightened by iteratively reintroducing the dropped constraints into the LP relaxation. One way of doing so is to add hyperplanes in order to separate the integer hull of the feasible set of the original problem from the optimal solution to the LP relaxation. Finding such cutting planes can however be very challenging. One can find numerous approaches in literature, some of which are of general purpose and can be applied to general mixed-integer problems (such as Gomory cuts [Gom60]), while others make use of the special problem structure (e.g. the first cutting planes introduced by Dantzig et. al. for the traveling salesman problem [DFJ54]). Another way of improving the dual bound by reintroducing integer constraints is branching. If the optimal solution of the LP relaxation is fractional and therefor not a feasible solution to the original problem, one can find an index i with i p ≤ such thatx ˆi R Z. We create two new subproblems by copying the original LP ∈ \ relaxation twice and adding the two additional constraints xi xˆi and xi xˆi , respectively. Each of the two new subproblems must then be solved≤ b c separately≥ d ande both objective values must be compared in order to find the gobal optimum. With this strategy, we can remove the fractional solution vector from the feasible sets of the subproblems, however at the price of creating two problems out of one. When solving the new subproblems, some of the variables with integer constraints can again become fractional, which leads to a rapidly growing tree of open subproblems, where the original LP relaxation without additional constraints is the root node. In order to find the best lower bound, which corresponds to the global optimal solution of the original MILP, one would have to solve all subproblems and compare their objective value. Luckily, in three cases further branching at a node is not necessary and therefor parts of the tree can be pruned:

If the corresponding LP of a node in the tree is infeasible, all descendants • are also infeasible. If the corresponding LP solution is feasible to the original MILP. • If the objective value of the corresponding LP of a node is greater-or-equal to • the objective value of the current best feasible solution to the original MILP.

The latter explains why employing heuristics to obtain good feasible solutions early is important. Both approaches of branching and cutting are combined in a method known as branch-and-cut which is the foundation of almost any successful MILP solver. Cut- ting planes are generated and used to tighten the LP relaxations in the tree. These can either be global cuts which are satisfied by all feasible integer solutions, or local cuts, that are only satisfied by all solutions from the currently considered subtree. If no more “useful” cuts are found, branching is carried out and the tree is pruned whenever possible.

8 3 Optimization of Technical Systems

The process of designing a technical system comprises the establishment of the layout, functionalities, as well as the used components. Empirical studies show that these initial decisions determine 70 to 80% of the total costs that occur during the total lifespan of the system [VDI03]. Besides investment and operation costs, maintenance and downtime costs during the expected planning horizon should be considered. This chapter provides an overview of optimization techniques that have been applied for the optimization of the design and operation of fluid systems in the context of discrete decisions.

3.1 Optimization of Fluid Systems

In recent years there have been a series of works that deal with the optimization of flow networks, e.g., with the optimization of water supply networks [MGM12]. The general mathematical problem underlying beneath the task of finding an op- timal system design and its optimal operation is challenging, not only because it contains both continuous and discrete variables. In addition, the networks can become large and temporal couplings over the planning period can occur. To find a reasonable compromise between model accuracy and solubility it is important to develop appropriate models and algorithms to solve these problems.

The optimal control task of a water network with continuous control variables is considered in [Kol12] where gradient-based optimization techniques are used to calculate the optimal control with fast algorithms. In general, the optimal operation of a water distribution network is a mixed-integer nonlinear program: Binary variables represent the switching of valves or pumps while the physical constraints and component characteristics are nonlinear and often nonconvex.

The problem of finding energy optimal settings of components within a network with a given layout has, e.g., been investigated in [Mar+12]. The authors use MILP techniques to solve control problems that include both continuous and dis- crete control variables. The model formulations contain nonlinearities and partial differential equations which are discretized and piecewise linearly approximated to make them accessible for MILP Algorithmics. The resulting MILP could be solved to global optimality. For solving operational planning tasks for a given network

9 3 Optimization of Technical Systems

sc. 1

sc. 2

sc. 3

sc. 4 FIRST STAGE FIRST VARIABLES

SECOND STAGE VARIABLES

Figure 3.1 – Problem Structure for a system synthesis problem. Binary first stage variables indicate the investment decisions, e.g. if a pump has been bought or not. Second stage variables for all load scenarios are coupled to the first stage variables. topology, decomposition methods like discrete dynamic programming, see, e.g., [ZS89], or Generalized Benders Decomposition [VA15] have been applied.

Besides the optimal control, also the optimal design of water distribution networks has been studied by several authors. It has to be noted that the literature in this context defines the term “design” in a special way. In the water distribution network community, the terms “design” and “layout” are used to distinguish two different optimization problems [DCS13]: While the layout problem deals with finding an optimal network topology, the design problem is to find the material and diameter of each pipe in a network with a given topology so that the total cost is minimized without violating any hydraulic constraints.

Traditionally, first, the layout of the pipe network is determined and taken as fixed during the process of component sizing [Raa11], p. 87. However, even the resulting pipe sizing problem was shown to be NP-hard [YTB84]. Several (meta-)heuristics like Simulated Annealing Tabu Search or Genetic algorithms have been applied to this problem, i.e., to the optimal assignment of discrete diameters to a given pipe topology. A good overview can be found in [DCS13]. Bragalli et al. [Bra+15] propose a solution method for the problem of finding the optimal size of pipe diameters of a water distribution network that uses a nonconvex continuous NLP (nonlinear programming) relaxation and a MINLP search.

Finding the optimal layout of a fluid system has been investigated by F¨ugenschuh et al. [FHM14] on the application example of sticky separation in waste paper processing. The authors present a MINLP for the simultaneous selection of the network’s topology and the optimal settings of each separator for the steady state.

10 3.2 Special Problem Structure of System Synthesis Problems

t = 1

t = 2

t = 3

t = 4 FIRST STAGE FIRST VARIABLES

SECOND STAGE VARIABLES

Figure 3.2 – Problem Structure for a system synthesis problem considering dy- namic processes. Binary first stage variables indicate the investment decisions. Second stage variables for all time steps are coupled to the first stage variables. Additional constraints link neighboring time steps (here blue colored)

Numerical results are obtained via different types of piecewise linearization of the nonlinearities and the use of MILP solvers.

The approach of piecewise linearization was also followed in [Alt+17a] where the optimal topology and settings for a ventilation system were found. Several load scenarios in steady state were considered. While the approach of linearization allows to use fast MILP solvers, it also introduces additional integer variables. Because of the special problem structure of system synthesis problems, see Section 3.2, additional integer variables have to be introduced for each load scenario that is considered. With the help of affinity laws derived from similarity theory, the amount of additional integer variables could be decreased.

3.2 Special Problem Structure of System Synthesis Problems

When modeling the optimization problem of finding the optimal system structure for a given load profile via MILP, a special problem structure arises: Binary so- called first stage variables indicate whether a component is part of the system or not. Whenever an optional component is not bought, it cannot be used to satisfy the required load scenarios. Therefore, all second stage variables describing the operation of the optional components must be coupled with the first-stage decision. This results in the problem structure that can be seen in Fig. 3.1.

11 3 Optimization of Technical Systems

The right hand side of this MILP contains different load requirements depending on the considered scenario. If one, e.g., considers two distinct load cases, the right side consists of two columns. The first stage variables which indicate whether a component is purchased or not take the same value in all load scenarios, as they represent decisions that have to be made once. In contrast, second stage variables can take different values in the different load cases and are dependent on the first stage variables. E.g., the volume flow through a pipe can only take a value unequal to zero, if this pipe is bought. For each scenario new columns with second stage variables are added. If second stage variables belong to different distinct load scenarios, they have no influence on each other and are not coupled. This results in the block structure as shown in Fig. 3.1. This structure changes if dynamic processes are investigated, e.g., by the consideration of accumulators or erosive wear. In this case, a block has to be added to the optimization model for each time step considered. The individual blocks in the matrix are coupled in between subsequent time steps, as are the dynamic processes, c.f. Fig. 3.2. The methods used in this thesis take advantage of this special problem structure when solving the related optimization problem.

12 4 Wear Processes and Algorithmic System Design

Given a specific load profile, it is often unclear which system structure is the optimal one, especially regarding expected costs due to failure and maintenance. Optimizing the layout and operation of a technical system can increase its lifespan considerably [VPP14]. Dispersion of load between multiple components might be favorable considering the expected time until the individual components have to be replaced. Thus, investing in more components than necessary can be beneficial regarding the availability of a system in between maintenance points. However, this also increases the complexity of the system. This can not only lead to higher investment costs, but also to additional effort in maintenance and operation, and to a higher probability for stochastic failures. In this chapter, an optimal system topology for a hydrostatic transmission system is found, taking into account the wear of its components and the resulting costs for replacements. The following formulation for the mathematical optimization model can also be found in the publications [Ver+16; Alt+15a; Alt+17b].

4.1 Mathematical Optimization Model for the Hydrostatic Transmission System

Finding the optimal design of the hydrostatic transmission system is a mixed- integer optimization problem consisting of two stages: First, find an investment decision in a set of up to four optional valves and their optimal topology. Secondly, find operating settings for the system given a specific load profile while taking into account the wear of each proportional valve. The quality of each topology option is determined by its ability to satisfy the load requirements and by the resulting wear of its components during operation.

According to the TOR approach the first step in finding an optimal system struc- ture is to define the required task. In this example, the hydrostatic transmission system has to satisfy two different load scenarios: One for moving its piston out- wards and one for moving it inwards, cf. Fig. 4.1(b). These two predefined scenar- ios are specified by velocity and force requirements for the piston. To satisfy these

13 4 Wear and Algorithmic System Design requirements, oil flow and pressure in the system have to be controlled by propor- tional and shift valves. For this application example we consider the construction kit in Fig. 4.1(a) of optional components to choose from. Available are up to four proportional and up to four shift valves and all their possible connections.

All possible designs for the hydrostatic transmission system are given by the com- plete graph G = (N,E). Its edges E represent the technical components of the system: one edge corresponds to the piston of the transmission system, while the other edges represent optional proportional and shift valves, and the possible pipes between them. The nodes N represent connection points between the compo- nents. Two distinct nodes s, t N represent a steady pressure source and the { } ∈ tank of the system. A binary first stage variable bi,j for each optional component (i, j) E indicates whether it is purchased (bi,j = 1) or not (bi,j = 0). The second stage∈ variables represent the system state during operation: All edges can convey a quasi-stationary flow Qi,j, whose sign determines the flow direction. A positive value of Qi,j represents flow from node i to node j, while a negative value repre- sents flow in the opposite direction. The nodes N of the graph can be regarded as measuring points with a certain pressure.

The graph is copied for each load case l L = 1, 2 : Based on the first stage ∈ { } decision made for the original graph G, graph G1 = (N1,E1) represents the op- eration of the purchased components when the piston is moving out, while graph G2 = (N2,E2) represents the operation for it moving in. Binary second stage vari- ables αl,vi,vj for each valve edge (vi, vj) in the set of valves l Gl allow to block purchased connections by shutting a valve during operation.V Pipe⊆ edges cannot be deactived.

Assuming incompressibility of the hydraulic fluid, we require the conservation of volume flow Q in each node n in each load case l:

n N , l L : Q = Q . (4.1) ∀ ∈ l ∀ ∈ l,i,n l,n,j (l,i,nX) ∈ El (l,n,jX) ∈ El

To enable the shutting of valves (v , v ) during operation in load case l, the i j ∈ Vl volume flow Ql,vi,vj is coupled with the binary open/shut indicator αl,vi,vj :

(v , v ) , l L : Q Q α , (4.2) ∀ i j ∈ Vl ∀ ∈ l,vi,vj ≤ max · l,vi,vj Q Q α . (4.3) l,vi,vj ≥ − max · l,vi,vj

Other physical relationships to be modeled are the hydraulic resistance laws. The pressure propagates along each edge, if components are connected:

(i, j) , l L : p p = ∆p . (4.4) ∀ ∈ El \Vl ∀ ∈ l,i − l,j i,j

14 4.1 Mathematical Optimization Model

PISTON 퐹표푢푡

푣표푢푡

퐹푖푛 CONSTRUCTION KIT OF 푣푖푛 DIFFERENT A B VALVES

P T

STEADY PRESSURE TANK SOURCE

(a) Design Problem: The construction kit consists of four proportional and four shift valves. Which combination results in the best system?

퐹

RUNNING-IN

FORCE TIME RUNNING-OUT

푣

RUNNING-IN

TIME VELOCITY RUNNING-OUT

(b) Load profile: The piston has to move outwards and inwards with a certain velocity against a specified force.

Figure 4.1 – Details on the hydrostatic transmission system. Figures taken from the joint publication [Alt+15b]

15 4 Wear and Algorithmic System Design

For closed valves, this propagation does not hold. We model this on/off constraint via the big-M formulation:

(v , v ) , l L : p p ∆p + p (1 α ), (4.5) ∀ i j ∈ Vl ∀ ∈ l,vi − l,vj ≤ vi,vj max · − l,vi,vj p p ∆p p (1 α ). (4.6) l,vi − l,vj ≥ vi,vj − max · − l,vi,vj

The pressure decrease ∆pi,j depends on the considered component (i, j). For open shift valves, ∆pi,j = 0 is assumed. For proportional valves, the resulting pressure decrease depends nonlinearly on the valve lift ul,vi,vj and on the volume flow Ql,vi,vj . This pressure decrease is described by a one-dimensional model:

2 ∆pl,vi,vj (vi, vj) prop,l, l L : Ql,vi,vj = · ζ d ul,vi,vj , (4.7) ∀ ∈ V ∀ ∈ s ρ · · · where prop,l is the set of proportional valves, ρ is the oil density, ζ is the pressure loss coefficient,V and d is the valve’s diameter. The system has to fulfill the load in each load case l L. The volume flow and the ∈ pressure decrease along the edge (pi, pj) representing the piston of the transmission system have to satisfy

l L : Q =v ˜ A , (4.8) ∀ ∈ l,pi,pj l · piston Fl pl,pi pl,pj = , (4.9) − Apiston wherev ˜l is the required velocity of the piston in load case l, Apiston is the area of the piston, and Fl is the required force in load case l. Regarding hydraulic valves, it has been found that the contamination of the hy- draulic fluid with solid particles is the most important cause of failure [Sch04].

According to [Alt+15a] the wear ∆wvi,vj of the control edge of a proportional valve (vi, vj) in load case l can be calculated by

1 ∆w = u exp t 1 . (4.10) l,vi,vj l,vi,vj · −3 ·Kl,vi,vj · l −     where = (∆p , u , u , d, ρ, c ) depends on operating parameters Kl,vi,vj K l,vi,vj l,vi,vj max V such as the pressure loss ∆p, the valve lift u, the maximum valve lift umax, the valve’s diameter d, the oil density ρ and the volume fraction of suspended solids cV . With this dependency, we can calculate the wear of each bought valve for a specific loading case l with length tl. To do so within the MILP, all of the above mentioned nonlinear equations were approximated by piecewise linear functions [VAN10]. For the optimal design of the hydrostatic transmission system one can imagine several objective functions, out of which some have been investigated in [Ver+16].

16 4.2 Extension of the Model

RUNNING OUT RUNNING IN

Figure 4.2 – Optimal topology if we maximize the run time of the system before the first component has to be replaced. Result of this objective is a system with hot redundancy. Load sharing allows to distribute the wear among the components. Figure taken from joint publication [Ver+16].

E.g, one could minimize the initial investment costs. In this case, the solution of the optimization problem will constitute a system consisting of the cheapest components that in principle satisfies the load requirements, however fails often due to worn out components that have to be replaced. Another reasonable approach is to maximize the expected failure free time of the system, i.e. the time until the first component has to be replaced. To do so, the following objective [Ver+16] can be considered:

min max ∆w(∆pl,vi,vj , ul,vi,vj , umax, d, ρ, cV ) . (4.11) (v ,v )∈V i j ∈ !! Xl L

The optimal solution corresponding to this approach, constitutes a complex system with multiple components and possibly hot redundancy, since the wear resulting from the load can be distributed among the components. Fig. 4.2 shows the solution for the considered application example.

4.2 Extension of the Model

When designing a technical system, one would like to prevent system failures. At the same time, the maintenance of the system should often ideally be limited to

17 4 Wear and Algorithmic System Design pre-defined periods. I.e., one wants the system to satisfy load requirements until the first maintenance point is reached, and worn or broken parts and components can be replaced. To make it possible to anticipate maintenance costs given such a maintenance strategy, the model formulation presented in the last section 4.1 was extended. The goal is to find an optimal system structure that balances two diverse objectives: An increase in availability by introducing redundancy can lead to a decreased maintenance effort due to wear. However, the system becomes more complex and more costly. If one wants to balance initial investment and maintenance costs during the ex- pected operating time of the system, several equations and second stage variables have to be added to the basic model. The first additional second stage variable ws,vi,vj is the wear of a valve (vi, vj) accumulated at the end of operation interval s. This accumulated wear can only be unequal to zero if the valve has been bought:

(v , v ) V, s = 2, ,T : w w b , (4.12) ∀ i j ∈ ∀ ··· s,vi,vj ≤ max · vi,vj where wmax is an upper bound for the accumulated wear. If the wear of a valve has reached this bound, the valve has to be replaced. The next additional equation describes the coupling of valve wear in between sub- sequent operation intervals:

(v , v ) V, s = 2, ,T : w = w − + ∆w , (4.13) ∀ i j ∈ ∀ ··· s,vi,vj s 1,vi,vj s,vi,vj where ∆ws,vi,vj the additional wear resulting from operating the valve in the current interval s. This additional wear in interval s is given by

1 ∆w = u exp t 1 . (4.14) s,i,j l,i,j · −3 ·Kl,i,j · l − XlinL    

To account for replacement costs, new second stage variables r 0, 1 are in- s,vi,vj ∈ { } troduced indicating if valve (vi, vj) is replaced at maintenance point s (rs,vi,vj = 1), or not (rs,vi,vj = 0). These second stage variables have to be coupled to the first stage variables. A valve can only be replaced, if it has been bought (bvi,vj = 1):

(v , v ) V, s = 1, ,T : r b . (4.15) ∀ i j ∈ ∀ ··· s,vi,vj ≤ vi,vj

If a valve (vi, vj) is replaced at the end of operating interval s, the wear accumulated until the end of this operating interval is set to zero:

(v , v ) V, s = 1, ,T : w w (1 r ). (4.16) ∀ i j ∈ ∀ ··· s,vi,vj ≤ max − s,vi,vj

18 4.2 Extension of the Model

In this case, Eq. 4.13 is no longer valid. Therefore, we replace it by an off/on- constraint via a big-M formulation:

(v , v ) V, s = 2, ,T : w w − + ∆w , ∀ i j ∈ ∀ ··· s,vi,vj ≤ s 1,vi,vj s,vi,vj (vi, vj) V, s = 2, ,T : ws,vi,vj ws−1,vi,vj + ∆ws,vi,vj wmax rs,vi,vj . ∀ ∈ ∀ ··· ≥ − · (4.17)

To find an optimal trade-off, the objective function of the extended optimization model now calculates the overall lifetime costs. It accounts for initial investment costs Cinvest and maintenance costs Cmaint due to replacements of worn out valves:

min z = Cinvest + Cmaint (4.18) = b C + r , (4.19) vi,vj · vi,vj s,vi,vj (vi,vjX)∈V (vi,vjX)∈V,s∈S where bvi,vj = 1 if a valve (vi, vj) is bought, and Cvi,vj is the price for this valve. rs,vi,vj is a binary variable indicating if valve (vi, vj) is replaced at the end of operation interval s.

When considering only the first part of this objective, the investment costs, the optimal solution corresponds to the smallest possible system built with the cheapest components, regardless of the downtime of the system because components have to be replaced. When considering only the second part of this objective, the costs for replacements of worn components, a very complex system with lots of hot redundancy evolves, since wear resulting from the load can be distributed among more components. The availability of the system increases, however as do the initial investment costs. Combining both parts allow us to find a trade-off between the two extremes.

Fig. 4.3 shows optimal topologies for the hydrostatic transmission system obtained with the above described objective function. Depending on the number of service points considered in the optimization model, the optimal system structure changes. When considering only one operating interval (i.e., we do not consider any service point and there is no possibility to replace components) the smallest possible system layout consisting of two proportional and two shift valves is chosen. This layout also corresponds to the layout with the lowest investment costs. The required load requirements during the operation interval are satisfied. The wear level of the valves does not reach the upper limit.

For two operating intervals we find a different optimal topology: the two shift valves are replaced by proportional valves even though they were considered twice as expensive as the switch valves. The wear can now be distributed among two proportional valves in each load case, a replacement of a valve at the maintenance point is not necessary.

19 4 Wear andOptimale Algorithmic Struktur System Design

■ Lastzyklen pro Wartungsintervall: 200 000

1 2 3 OPERATING INTERVALS

2. Juli 2015 | Prof. Dr.-Ing- Peter Pelz| Auslegung eines hydrostatischen Getriebes | Lucas Farnetane 14 Figure 4.3 – Optimal system structures for the hydrostatic transmission system. Depending on the number of operating intervals considered a differ- ent optimal system structure is found. Figure taken from the joint publication [Alt+15b]

If a third operation interval is considered, an optimal topology option consists of three proportional and two shift valves. In contrast to the smallest possible system consisting of two switch and two proportional valves, during the outwards- movement of the piston the throttling is executed via a parallel connection of two proportional valves instead one. This hot spare allows to distribute the wear among two components. As a result, the lifespan of these two valves is made more compatible with the time in between service points. Yet, for an extended operation of three operation intervals, three replacements due to worn off components have to be made: One of the proportional valves in the parallel connection has to be replaced once. The single proportional valve on the left has to be replaced twice. Since it throttles the pressure during the inward movement of the piston and their is no hot redundancy, it suffers from a high wear. One of the other two proportional valves has to be replaced one time. The presented optimization model allows to find an optimal system structure for a specific technical system with a maintenance plan, given deterministic wear of the components. It can in principle be applied to other technical systems, since boundary conditions, like the number of loading cycles per interval, can be adapted. For the consideration of further service points, new second stage variables have to be added to the model. The worst case runtime for solving such an optimization model increases exponentially with the number of operating intervals. While this chapter focused on modeling aspects, the next chapters show how decomposition techniques can be applied to solve the problem for a larger number of service intervals.

20 5 Topology Optimization

5.1 Superstructures of Possible Layouts

Optimizing the topology of a technical system requires to describe all possible topology options. This superstructure of possible layouts can be represented by a complete graph G = (E,V ) where each vertex v in the set of vertices V corresponds to an optional component and each edge in the set of edges E to an optional connection between two optional components. As can be seen in Tables 5.1 and 5.2, the number of possible system topologies increases sharply with the number of optional components. This is especially the case for the application example of the hydrostatic transmission system where we consider a set of eight optional components. Moreover, the number of possible subgraphs cannot be restricted to series-parallel subgraphs for this system, but can contain cycles. The combinatorics resulting from the complete graph of all system structures make the problem of finding an optimal topology and operation strategy difficult to manage. In this work, we follow an approach that separates the investment decision from the subsequent operation decision. I.e., first, we find a good proposal for the first stage variables that represent if components are bought or not and how they are connected. Secondly, we find a good operation strategy for this topology option. Fig. 5.1 shows exemplary system topologies for the hydrostatic transmission sys- tem. The task of the topology optimization is to find the optimal layout of valves, i.e., how many should be bought and how should they be connected.

5.2 Number of Possible Topology Options

All possible topology options can be represented by one “supergraph” that com- bines all the graphs representing single topology options. This supergraph depends on the specific optimization problem: for the case of booster stations, it can be reasonable to restrict it to series-parallel connections of a given set of optional pumps. The number of series-parallel graphs for n components is shown in Table 5.1. One can see, that even for this special case of only series-parallel connections,

21 5 Topology Optimization

Number of nodes Number of graphs 1 1 2 2 3 8 4 52 5 472 6 5504 7 78416 8 1320064 9 25637824 10 564275648 11 13879795712 12 377332365568 13 11234698041088 14 363581406419456 15 12707452084972544

Table 5.1 – Number of series-parallel graphs with n labeled edges. Also called yoke- chains by Cayley and MacMahon [Slo]. In this representation, each component of the fluid system is represented by a labeled edge. the number of possible topology options still grows rapidly with the number of components.

For the case of the hydrostatic transmission system, a restriction to only series- parallel networks is not possible. The number of possible topology options for this case can be deduced as follows: Given a set of n optional distinguishable components (valves), the piston, a sink (tank) and a source (pump). The number of possible subsystems that can be built of n optional components and of the piston equals the number of connected labeled graphs with n + 1 nodes which is given in Table 5.2. Each of these subsystems can be connected to the source and the sink in different ways. While a direct connection of the piston to the tank or the source is not considered, the other n nodes can be connected to the source and the sink as follows:

n • Connect n 1 components to the source and 1 to the sink. There are 1 = n choices for− the component that is connected to the sink.
n • Connect n 2 components to the source and 2 to the sink. There are 2 choices for− the component that is connected to the sink.
n • Connect n 3 components to the source and 3 to the sink. There are 3 choices for− the component connected to the sink.
• ...

22 5.2 Number of Possible Topology Options

Number of nodes Number of graphs 1 1 2 1 3 4 4 38 5 728 6 26704 7 1866256 8 251548592 9 66296291072 10 34496488594816 11 35641657548953344 12 73354596206766622208 13 301272202649664088951808 14 2471648811030443735290891264 15 40527680937730480234609755344896

Table 5.2 – Number of connected graphs with n labeled nodes, c.f. [Bon15]. Each node represents one component of the fluid system.

The number of possible choices to connect k out of n components to the sink equals n the number of k-combinations, i.e., k . The number of possibilities to partition the set of components into a set of components connected to the sink and another
set connected to the source can be computed by the Stirling number of the second kind. It yields the number of ways to partition a set of n objects into k non-empty n subsets and is denoted by k . For k = 2, it can be calculated via the explicit formula [Sha68] { }

n 1 2 2 = ( 1)2−j jn. (5.1) 2 2! − j j=0 n o X   With this relation the number of possible topology options for the application example of the hydrostatic transmission system can be computed. Given eight optional components and the piston, we have 66296291072 possible graphs, cf. Table 5.2, that can be connected to source and sink in

8 1 2 2 = ( 1)2−j j8 = 127 (5.2) 2 2! − j j=0   X   different ways. This accounts to 66296291072 127 = 8419628966144 possible sys- tems. An enumeration of all of them seems hardly· like a suitable way to determine the optimal one.

23 5 Topology Optimization

Figure 5.1 – Exemplary system topologies for a hydrostatic transmission system consisting of different connections of proportional and switch valves

5.3 Topology Options for the Hydrostatic Transmission System

The graph in Fig. 5.4 shows the undirected graph of all possible connections in between optional components of the hydrostatic transmission system considered in this work. Series connections must only be considered for proportional valves: While throttling the pressure via two or more proportional valves might be bene- ficial, a series connection of switch valves is not reasonable.

In this exemplary case the set of optional components consists of four proportional and four switch valves out of which up to four can be chosen, respectively. Fig. 5.2 shows runtimes of the commercial solver IBM ILOG CPLEX for the problem of finding an optimal system structure, given specific load requirements and a number of service intervals. Whenever the run time exceeded 24 hours, the computation was stopped and the best solution found so far was saved. As can be seen, the run times are in the order of hours even for small instances. Many of the problems could not be solved within 24 hours at all. Therefore, this work investigates the development of efficient primal heuristics for determining the system structure.

As can be seen in Fig. 5.1 possible system layouts for the hydrostatic transmission system are not necessarily series-parallel, c.f. Section 5.3.1, but can contain cycles. Therefore, the generation of only series-parallel networks in a primal heuristic like in [Utz14] is not possible. However, if one looks at each of the two loading cases of moving the piston in and out separately, hydraulic fluid flowing in cycles within one load case is not desirable and a series-parallel topology is favored. This is achieved in reality by blocking specific edges via the proportional and shift valves during each of the two loading cases, cf. Fig. 5.3. This results in a series-parallel

24 5.3 Topology Options for the Hydrostatic Transmission System

TOPT TFEAS 105 TIME LIMIT EXCEEDED

104

103 RUNTIME in s 102

1 2 3 4 5 7 9 11 # OPERATION INTERVALS

Figure 5.2 – Runtimes of the commercial solver IBM ILOG CPLEX for the problem of finding an optimal system structure for the hydrostatic transmis- sion system consisting of up to eight valves. Given is a specific load requirement and a number of service intervals. The runtime until a first primal solution is found, is denoted by TFEAS and the runtime until the global optimal solution is found is denoted by TOPT. For bars that reach the time limit, no solution could be found within 24 hours. The computations were executed at the Lichtenberg High Per- formance Computer of TU Darmstadt, by using Intel Xeon compute nodes x86-64 (AVX2) with a base frequency of 2.5 GHz, and CPLEX version 12.6.1 with up to 4 threads.

25 5 Topology Optimization system of open valves for each loading case. This property is used in the developed primal heuristic. When finding good primal solutions to the valve topology the idea is as follows: 1.) Find a good series-parallel layout of proportional valves for each loading case. 2.) Combine the two graphs. 3.) Add shift valves where necessary. To guarantee that each load case can be satisfied with the combined graph, each path from the pump to the piston and each path from the piston to the tank must be blockable, even though some of the proportional valves may be open during both load cases of moving in and moving out. The algorithm to determine where to put additional shift valves is as follows:

Data: Two series-parallel graphs G1 = (V1,E1),G2 = (V2,E2) for loading cases l1 and l2. Components are represented by the graphs’ nodes, connections between the components are represented by the graphs’ edges. PV1 V1 and PV2 V2 are the sets of nodes that are proportional⊆ valves and PV⊆ is their union. Result: Valid joined graph G. Join set of nodes and set of edges: V = V1 V2, E = E1 E2; for each edge (i, j) E do ∪ ∪ ∈ if ((i, j) / E1 E2) and (i, j / PV (PV1 PV2)) then add shift∈ valve∩ sv: V := V ∈ sv \; ∩ add corresponding edges: E∪:= { E} (i, sv), (sv, j) ; remove obsolete edge: E := E (∪i, {j) ; } \{ } end end Algorithm 1: Building one valid graph from two series-parallel graphs. The two series-parallel graphs for each loading case consist themselves of two series- parallel graphs connected in series to the piston, cf. Fig. 5.3. MacMahon [Mac90] showed that these series-parallel graphs can be represented by so-called series- reduced trees which will be outlined in the next chapter.

5.3.1 Series-Parallel Networks

Often, not all subgraphs of the complete graph of optional components are valid solutions for the system structure. If one can restrict the optimization problem to a special kind of graph structure, the resulting combinatorics become more manageable. The idea in the developed primal heuristic is to find series parallel graphs for each load case, and combine them to a valid system for both load cases. Series parallel graphs are a special class of graphs that can be used to model certain types of electric networks [Duf65]. They have attained special interest, because a number of standard graph problems that are NP-complete for general graphs are solvable in linear time on series parallel graphs [TNS82].

26 5.3 Topology Options for the Hydrostatic Transmission System

RUNNING OUT

P K P

2

TREE 1 TREE 2

RUNNING IN

P K P

2 3 4 TREE 3 TREE 4

Figure 5.3 – The graph of the hydrostatic transmission system is not series-parallel. However, for the two distinct load cases of moving the piston in and out, a series-parallel topology is achieved via the blocking of valves. While proportional valves serve to throttle the pressure, the job of the available switch valves is to make sure that no unwanted flows of the hydrostatic fluid occur. If we consider for each load case only open valves, the resulting network for a single load case can be represented by a serial connection of a series-parallel network of valves, the piston and another series-parallel network. The single series parallel net- works are represented by series reduced trees in the developed primal heuristic. Four trees represent a topology option that is combined to a valid system structure by Algorithm 1.

27 5 Topology Optimization

Figure 5.4 – Graph representing superstructure of all considered system layouts for the application example of the hydrostatic transmission system. Possible are multiple different connections of up to eight valves.

Series parallel graphs can be defined recursively. The following definition is taken from [Epp92].

Definition 1. A directed graph G is two-terminal series parallel, with terminals s˜ and t˜, if it can be produced by a sequence of the following operations:

1) Create a new graph, consisting of a single edge directed from s˜ to t˜.

2) Given two two-terminal series parallel graphs GX and GY , with terminals s˜X , t˜X , s˜Y , and t˜Y , form a new graph G = P (GX ,GY ) by identifying s˜ =s ˜X =s ˜Y and t˜= t˜X = t˜Y . This is known as the parallel composition of X and Y.

3) Given two two-terminal series parallel graphs GX and GY , with terminals s˜X , t˜X , s˜Y , and t˜Y , form a new graph G = S(GX ,GY ) by identifying s˜ =s ˜X , t˜X =s ˜Y , and t˜ = t˜Y . This is known as the series composition of GX and GY . An undirected graph is two terminal series parallel with terminals s˜ and t˜ if for some orientation of its edges it forms a directed two terminal series parallel graph with those terminals.

Definition 1 shows that series parallel graphs can be constructed recursively. Find- ing the number of series-parallel networks that can be formed using a given number of components, is a known problem in combinatorial mathematics based on the

28 TABLE I. Two-terminal series-parallel networks5.4 for Localn =1, 2 Search, 3, 4. On for each System row, the Topologies two net- works are dual. The an’s (number of networks) are 1, 2, 4, 10, . . . The bn’s (number of essentially series or, by duality, essentially parallel networks) are 1,1,2,5,... number of essentially essentially number of elements series parallel networks 1 X1 X1 1

2 X1X2 X1 + X2 2

3 X1X2X3 X1 + X2 + X3 4

X1(X2 + X3) X1 + X2X3

4 X1X2X3X4 X1 + X2 + X3 + X4 10

X1X2(X3 + X4) X1 + X2 + X3X4

X1(X2 + X3 + X4) X1 + X2X3X4

X1(X2 + X3X4) X1 + X2(X3 + X4)

(X1 + X2)(X3 + X4) X1X2 + X3X4

Figure 5.5 – Number of series-parallel networks of order 1 to 4. In this represen- an =2bn δn,1. (1) tation, all edges correspond− to unlabeled components, like pipes or pumps. MacMahon called a series connection a “chain” and a parallel connection a “yoke”[Mac90]. A series-parallel network can be con- III. ENUMERATION AND RECURRENCE GENERATING FUNCTION structed recursively by series and parallel connections of chains and In this section,yokes. we Byexplain interchanging how to enumerate the series the and different parallel net connections,works, i.e. compute the series- the a or b .Byuseoftheduality,weconsideronlyaparallel network in each row yieldsσ a-network conjugateS series-parallelwith n 2elements.It network. n n n ≥ is composed withFig.kπ taken-structures from of [Gol97].w ,w ...,w elements, with k 2and w = n.Its 1 2 k ≥ i i boolean representation is Sn = Pw1 Pw2 ...Pwk .Bygroupingtogethertheequal! wi,wecan r1 r2 rq use a symbolic exponential notation Sn = P1 P2 ...Pq . workThe of MacMahonrj are non-negative in 1890, (r seej 0) [Mac90]. integers ; MacMahon they form a partition showedof hown with to at generate least two all { } ≥ r1 r2 rq seriesparts parallel : n = graphsj jrj is partitioned with n components in k = j rj byparts. representing We note a partition them by as series 1 2 ...q reduced. For example, the partitions of 5 are 15, 132, 123, 122, 14, 23 . trees, i.e., trees! without internal nodes{ ! of degree 2. Each} leaf in such a tree repre- Then, to obtain bn (the number of σ-network with n elements), we first must compute sentsthe contributionthe minimal of aseries given parallelpartition, network and finally, of we only will suonemoverallthepartitions.Thanks component, while the other nodesto the represent above decomposition, a series or the parallel number connection of S with a ofgiven networks. partition Eachr is a tree product represents of as n { j} twomany conjugated factors as networks:j,dependentonlyon one thatb isj and essentiallyrj. parallel and one that is essentially serial,For c.f. example, Fig. 5.5. it is obvious that, for b5,thecontributionofthepartition(23)isb2b3, because b2 is the number of choices for the first π-structure of two elements, and b3 for the second of three elements. When two π-structures have the same number of elements, i.e. rj =2,theircontribution b2 b b / 5.4is not Localj but, to removeSearch duplications for System obtained by permutations, Topologiesj( j +1) 2. More generally, for arbitrary r 0, the factor is j ≥ In this section a primal heuristic for finding good topology options for the hydro- static transmission system is presented. 3

The employed local search is a heuristic for tackling computationally hard opti- mization problems. Local search algorithms are widely applied to problems from mathematics, computer science, operations research, engineering, and bioinformat- ics. They have been successfully applied to NP-hard combinatorial optimization problems for many years. A prominent example is the 2-opt algorithm for the Traveling Salesman Problem [Cro58].

29 5 Topology Optimization

Regarding heuristic algorithms, there are two categories: constructive algorithms and local search algorithms [MAK07]. While a constructive algorithm creates a solution step by step and obtains the complete solution only at the last step, a local search algorithm pursues a different strategy: Local search algorithms traverse the space of all possible solutions of a problem, the so-called search space, in order to find high quality solutions. Starting from one solution (e.g. known or found by a heuristic), local search iteratively moves to a “neighboring” solution by applying local changes. If a found solution becomes the new starting point for the next iteration is determined by a previously established acceptance criterion. The search process is terminated if a time or iteration limit is reached or a solution is deemed optimal. A common choice is to terminate when the best solution found so far has not been improved in a given number of steps. In contrast to constructive algorithms, local search is an anytime algorithm: being interrupted at any time, it can return a valid solution. When implementing a local search algorithm for the hydrostatic transmission sys- tem, we resort to problem-specific knowledge. In order to find valid series-parallel connections of the available set of valves, we make use of their representation by series-reduced trees which was shown in Section 5.3.1. Based on this representa- tion, and the combination of two series-parallel graphs for each laoding case it is possible to elegantly traverse the search space of candidate system structures. The following definitions in the context of local search are taken (with small changes) from [MAK07]. Definition 2. An instance of a combinatorial optimization problem is a pair (S, f), where the solution space S is a finite or countably infinite set of solutions and the cost function f is a mapping f : S R that assigns a real value to each solution in S called the cost of the solution. → Definition 3. A combinatorial optimization problem Π is specified by a set of problem instances and it is either a minimization or a maximization problem. The problem is to find for a given instance (S, f) a solution s∗ S that is globally optimal. For a minimization problem, this means that f(s∗) ∈f(s) has to hold for all s S. ≤ ∈ In order to traverse the solution space from one solution to the next, we need to define all solutions that are near the given one. This is done by the so-called neighborhood function. Definition 4. For an instance (S, f) of a combinatorial optimization problem, a neighborhood function is a mapping N : S (S), where denotes the powerset of S. The neighborhood function specifies→ for P each solutionP s S a set N(s) S, which is called the neighborhood of s. The cardinality of ∈N(s) is called the⊆ neighborhoodsize of s. We say that a solution sˆ is a neighbor of s if sˆ N(s). The neighborhood function is symmetric if sˆ N(s) s N(ˆs). ∈ ∈ ⇔ ∈

30 5.4 Local Search for System Topologies

The process of continually moving from one solution to its neighbors can be mod- eled as a walk through the neighborhood graph.

Definition 5. The neighborhood graph of an instance (S, f) of a combinatorial op- timization problem and an accompanying neighborhood function is a directed node- weighted graph G = (V,E). The node set V is given by the set S of solutions. The arc set E is defined such that (i, j) E iff j N(i). The weight of a node is given by the cost of the corresponding solution.∈ If the∈ neighborhood function is symmetric the directed graph can be simplified to an undirected graph by replacing arcs (i, j) and (j, i) by a single edge i,j . { }

5.4.1 Neighborhood Function

In order to walk from one valid topology option to the next one, we have to define a neighborhood function. Instead of implementing it on the graph representing the network, we operate on the respective series-reduced tree representation. We define the neighborhood function as a combination of neighborhood functions operating on the trees representing the single series parallel graphs and one neighborhood function that allows to switch the trees for the two load cases. The neighborhood functions for each tree are:

Swapping components Starting from the series-reduced tree representation • of a valid topology option, a permutation of the leaves is executed, cf. Fig. 5.6. If leaves have the same parent, their permutation is considered to yield the same system and is therefore not considered. This neighborhood is called the swap neighborhood NSWAP. Replacing components Starting from the series-reduced tree representa- • tion of a valid topology option, the leaf representing a component c1 is re- placed by a leaf representing component c2, cf. Fig. 5.7. This neighborhood is called the replace neighborhood NREPLACE. Adding components Starting from the series-reduced tree representation • of a valid topology option, a component c is added to the network by adding a leaf representing this component. If a leaf is added to a parent that has no children, two leaves are added and the parent becomes the sibling of the added leaf, cf. Fig. 5.9. This neighborhood is called the add neighborhood NADD Deleting components Starting from the series-reduced tree representation • of a valid topology option, a component c is deleted from the network by deleting its corresponding leaf. Special cases occur if the leaf to be deleted has only one sibling, cf. Fig. 5.10. If this leaf is the only leaf on his level, two different operations are executed depending on its parent. If its parent is the

31 5 Topology Optimization

S S

1 1 A) 3 2 3 2 2 3 1 2 1 3

S S 1 3 B) 3 1 3 1 2 2 1 2 3 2

Figure 5.6 – Exemplary executions of the swap neighborhood function. Starting from a valid tree, the neighborhood function NSWAP swaps two of its leaves. The resulting topology of the series-parallel graph is shown on the right. Leaves that have the same parents are not swapped since the resulting topology option is merely a permutation of components that are connected in parallel or in series and this swap option is considered equally good.

root, the root, the leaf and the connecting edge are deleted. Subsequently the interpretation of the tree is switched from series to parallel and vice versa. If its parent is not the root, the parent, the leaf, the other children of the parent and all edges from the parent to his children are deleted. The grandchildren of the parent are adopted by the parent of the parent, c.f. Fig. 5.10. If the leaf has only one sibling that is also a leaf, both leaves are deleted, see also Fig. 5.10. This neighborhood is called the delete neighborhood NDELETE Switch interpretation Starting from the series-reduced tree representation • of a valid topology option, the tree is conjugated. If it was interpreted as essentially series it is interpreted as essentially parallel and vice versa, see Fig.5.10. This neighborhood is called the switch interpretation neighborhood NSWITCH

In order to combine the individual trees to a valid forest, the following requirements must be met when executing a neighborhood function:

A component can only replace a component in tree T if it is not part of tree • 1 T2 and vice versa. A component can only replaced a component in tree T3 if it is not part of tree T4 and vice versa. A component can only be added to tree T if it is not part of tree T and vice • 1 2 versa. A component can only be added to tree T3 if it is not part of tree T4 and vice versa.

32 5.4 Local Search for System Topologies

S S 1 2 3 1 2 4 1 2 3 1 2 4

Figure 5.7 – Exemplary executions of the replace neighborhood function. Starting from a valid topology option, components can be replaced with the neighborhood function NREPLACE. In the series-reduced tree repre- sentation on the left, a leaf representing one component is replaced by a leaf representing another component.

Another neighborhood function operates on the forest: Swapping Trees Starting from a valid representation of the system as a • forest of four trees as forest, cf. Fig. 5.3, trees T1 and T3, and T2 and T4 are swapped. This neighborhood is called the swap trees neighborhood NSWAP TREE. The complete neighborhood function is given by N = N N N N N N . (5.3) SW AP ∪ REP LACE ∪ ADD ∪ DELET E ∪ SWITCH ∪ SW AP T REE It has to be shown that from every solution consisting of a forest with n components, every solution with 1 c n + 1, c Z components can be reached. For single trees, this is easy to deduce.≤ ≤ For the case∈ of a series parallel graph with unlabeled nodes MacMahon [Mac90] showed that by adding and removing leaves in the way described in the neighborhood functions NADD and NDELETE, all series parallel graphs that are either essentially series or essentially parallel can be generated recursively from a tree with one leaf. This allows us to reach solutions with a different number of components. For labeled components it remains to be shown that all permutations of these components on the trees can be reached. This can be achieved by the swap and replace neighborhoods. The neighborhood function described here can only add a leaf to a tree if the corresponding component is not already part of the second tree of the same load case. In this case, the component has to be deleted from the second tree before being added to the first tree. This is possible by a combination of NADD and NDELETE operating on the two different trees of one load case.

The neighborhood function NSWAP TREES allows to swap the topologies found for load case one and load case two. These topologies could also be swapped by deleting components from the trees of one load case and adding them to the trees of the other load case. However, the multiple execution of the delete function can lead to infeasible topology options. Therefore, an additional function to swap the trees of the load cases was implemented in the heuristic.

33 5 Topology Optimization

S P 1 A) 1 2 1 2 1 2 2

S P 1 1 2 B) 3 3 3 2 3 1 2 1 2

Figure 5.8 – Exemplary executions of the switch neighborhood function. Starting from the series-reduced tree representation of a valid topology option, the interpretation of a tree is switched from essentially series to essen- tially parallel and vice versa.

S S A) 1 2 1 2 3 1 2 1 2 3

S S 1 1 2 1 2 2 B) 2 3

1 3

Figure 5.9 – Exemplary executions of the add neighborhood function. Starting from the series-reduced tree representation of a valid topology option, the neighborhood function NADD adds a leaf. In the graph representation on the right this corresponds to adding a component to the already existing topology. This new component can be added to a arbitrary place in the existing topology and can be connected in series, cf. A), or in parallel, cf. B). Only components that are not already used in the corresponding load case are added.

34 5.4 Local Search for System Topologies

S S A) 1 2 3 1 2 1 2 3 1 2

S S 1 B) 3 1 3 3 1 3 2 1 2

S

4 P C) 1 2 1 2 3 3 4 3 3 1 2 1 2

S

4 S D) 1 2 3 1 2 4 1 2 4 4 3 1 2

Figure 5.10 – Exemplary executions of the delete neighborhood function. Starting from the tree representation of a valid topology option, the neigh- borhood function NDELETE allows to delete an arbitrary leaf. This corresponds to the deletion of a component at an arbitrary position in the network graph shown on the right. If the leaf has only one sib- ling that is also a leaf, both leaves are deleted, cf. B). Special cases also occur if the leaf to be deleted has only one sibling, cf. C), D). If the parent is the root, the root, the leaf and the connecting edge are deleted, cf. C), and the interpretation of the tree is switched from series to parallel and vice versa. Otherwise, in case D), the parent, the leaf, the other children of the parent and all edges from the par- ent to his children are deleted. The grandchildren of the parent are adopted by the parent of the parent.

35 5 Topology Optimization 5.5 Simulated Annealing

Local search algorithms are able to quickly find a locally optimal solution by itera- tively improving the current solution. Aiming for a global optimal solution, one can apply meta-heuristics that prevent the algorithm to stop at the first encountered local optimum. According to [MAK07] these meta-heuristics can be classified into two categories: those performing multiple walks in the neighborhood graph, and those performing a single long walk while using a mechanism to escape from local optima. One popular approach for the latter is Simulated Annealing.

This meta-heuristic is inspired by the annealing process in condensed matter physics, i.e., the evolution of a solid in a heat bath to thermal equilibrium. The process starts by heating the solid until thermal stresses are released and the particles are arranged randomly in a liquid state. By slowly decreasing the temperature, the solid reaches thermal equilibrium: crystals emerge which correspond to the energetically most favorable states.

Based on this analogy, solutions to an optimization problem correspond to states of a system of particles, while the cost of a solution is given by the energy of the corresponding state. At thermal equilibrium, the probability of the system being in a certain state j with energy Ej is given by the Boltzmann distribution [Gar95]

exp( Ej ) p(E ) = − kBT , (5.4) j C with N C = exp ( E /k T ) (5.5) − i B i=1 X representing the set of all possible states, where kB is the Boltzmann constant, T is the temperature of the system and N is the total number of system states. The process of annealing can be simulated by the Metropolis algorithm[Met+53]: Starting at time t and state s, a candidate state s0 for the next time step t + 1 is generated randomly. This new candidate is accepted or rejected based on the difference between the energies of the states s and s0. The state s0 is accepted if

0 p E 0 E p = s = exp s − s 1. (5.6) p − k T ≥ s  B  If we apply the Metropolis algorithm to a discrete optimization problem this means that starting from a solution s, the solution s0 is accepted as next solution at iteration k with an acceptance probability given by

36 5.5 Simulated Annealing

1 iff(s0) f(s) 0 p(s ) = 0 ≤ . (5.7) exp f(s)−f(s ) iff(s0) f(s) ( c(k) ≥   that depends on the objective value f(s) of this solution and on the parameter c(k). c(k) is called control parameter or temperature and gives the expected value of the threshold in iteration k [MAK07].

f(s)−f(s0) This can be implemented by comparing the value exp c(k) with a random number generated from the uniform distribution on the interval [0, 1) [MAK07], cf. Algorithm 4. Solution candidates are selected randomly from the neighborhood.

Data: some intial solution sinitial Result: solution sbest with f(sbest) f(sinitial) begin ≤ s := some initial solution; k := 1; sbest := s; while not stop criterion do generate an s0 N(s); if f(s0) f(s)∈then s := ≤s0 else 0 if f(s)−f(s ) then exp c(k) > random[0, 1) 0 s := s  end end k := k + 1; if f(s) f(sbest) then sbest≤:= s end end end

Algorithm 2: Simulated annealing algorithm for a minimization problem.

In [GL93] theoretical aspects of simulated annealing are shown: as long as it is possible to move from a solution s to any other solution s0 with non-zero proba- bility, simulated annealing converges for a given temperature towards a stationary distribution that is independent from the initial solution. As the temperature goes to zero during the cooling process, the form of the stationary distribution converges to a uniform distribution over the set of optimal solutions. In this work, for the (k) 100 temperature or control parameter in iteration k, c = 1+k is chosen.

37 5 Topology Optimization 5.6 Implementation and Benchmarks

For initialization, the simulated annealing algorithm is provided with an intial layout for the hydrostatic transmission system. New candidates are generated via the neighborhood functions described in Section 5.4.1. One of the neighboorhood functions is chosen randomly and randomly applied to one of the trees. For the function NSW AP two randomly chosen nodes of the tree are chosen. If one of the neighborhood functions NDELET E and NREP LACE is executed, one randomly chosen node of the tree is deleted and replaced, respectively. For the neighborhood function NADD only the set of nodes that are not already part of the respective load case is considered. Fig. 5.11 shows the solutions obtained via simulated annealing. For two small instances with few service intervals, the commercial solver CPLEX is able to find global optimal solutions in acceptable time, outperforming the simulated anneal- ing heuristic. However, when considering more realistic instances of 3 operation intervals and more, CPLEX is in many cases unable to find the global optimal or even a primal solution within 24 hours. Table 5.3 shows the objective values obtained via simulated annealing. For the smallest number of operation intervals, the algorithm cannot find the global optimal solution found by cplex. For three operation intervals, cplex does find a feasible, but not global optimal solution and a lower bound of 4.0. Via simulated annealing a solution with the same objective value of 9.0 is found as well.

30

20

SIM. ANN. DUAL BOUND

OBJECTIVE 10

0 0.5 1 1.5 2 RUNTIME in s 105 · Figure 5.11 – Progress of the solution quality obtained via simulated annealing for 10 operation intervals. There is a huge gap to the dual bound found by cplex. Cplex is not able to find a feasible solution during the runtime of the benchmark. The progress of the curve shows nicely the ability of simulated annealing to overcome local minima.

38 5.6 Implementation and Benchmarks

SIM ANN. OPT. VALUE 12

10 OBJECTIVE

0 0.5 1 1.5 RUNTIME in s 105 · Figure 5.12 – Results for the simulated annealing heuristic for the case of three operation intervals. For this smaller instance, an optimal solution can be obtained with the commercial solver cplex. The primal heuristic yields an upper bound that exceeds the optimal objective value by 1.0.

intervals zSim.Ann. zprimal zdual 2 7.0 5.0 5.0000 3 9.0 9.0 4.0000 4 10.0 – 4.0000 5 12.0 – 4.0000 6 17.0 – 3,7615 7 18.0 – 4.0000 10 25.0 – 3.3122

Table 5.3 – Best objective values zSim.Ann. obtained by three runs of the simulated annealing heuristic, each run was 24 hours. This objective value is compared to the best primal solution and the dual bound found by the commercial solver cplex. “–” marks that cplex was not able to find a feasible solution within 24 hours. The computations were executed at the Lichtenberg High Performance Computer of TU Darmstadt, by using Intel Xeon compute nodes x86-64 (AVX2) with a base frequency of 2.5 GHz and CPLEX version 12.6.1 with up to 4 threads.

39 5 Topology Optimization

40 6 Dynamic Programming

Dynamic programming (also known as dynamic optimization) is a method for solv- ing complex problems by breaking them down into simpler subproblems. It is often applied to discrete optimization problems for which a sequence of optimal decisions has to be made. In this chapter, we will present how Dynamic Programming can be applied to find the optimal replacement strategy for a hydrostatic transmission system whose components are prone to wear.

6.1 Fundamentals

Dynamic programming is a promising approach if the considered optimization prob- lem consists of many similar subproblems and an optimal solution can be con- structed of optimal solutions to these subproblems. Key feature of this approach is that optimal solutions to larger subproblems are also always optimal solutions to the subproblems that they contain. This is expressed by Bellmans optimality principle:

An optimal policy has the property that whatever the initial state and initial ” decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.“ [Bel57]

The optimality principle states that optimal policies have optimal subpolicies. That this principle holds can be verified by the following consideration [LM06]: If a policy has a subpolicy that is not optimal the replacement of this subpolicy by an optimal one would improve the original policy.

To solve a problem via Dynamic Programming, solutions to the smallest subprob- lems are directly calculated and suitably assembled to a solution for the next larger subproblems. This process of extensioning the problem is repeated until the origi- nal problem has been solved. Once calculated partial results are stored in a table. This allows to re-use temporary solutions in subsequent calculations of similar subproblems and thus avoids costly recalculation of the same problems.

41 6 Dynamic Programming 6.2 Application to Dynamic Decision Problems

A problem can be solved via Dynamic Programming if it is characterized by the following features [Hil07]:

1. The problem can be divided into stages – within each of these stages a policy decision is required. 2. The beginning of each stage is associated with a number of states. 3. The policy decision at each stage transforms the current state to a state asso- ciated with the beginning of the next stage. To find an optimal policy for the overall problem, a solution procedure can be designed – it prescripts the optimal policy decision at each stage for each of the possible states. 4. Given the current state, an optimal policy for the remaining stages is indepen- dent of the policy decisions applied in previous stages. 5. The solution procedure begins by prescribing the optimal policy decision for each of the possible states at the last stage. 6. A recursive relationship that identifies the optimal policy for stage n, given the optimal policy for stage n + 1, is available (e.g.):

∗ fn(sn) = min fn(sn, xn) , (6.1) xn { } w.r.t. N: Number of stages, • n: Label for current stage, • s : Current state for stage n, • n S : Set of reachable states for stage n, • n x : Decision variable for stage n, • n x∗ : Optimal value of x , • n n X : Set of decisions for stage n, • n f (s , x ): Contribution of stages n, n + 1,...,N to objective function if • n n n system starts in state sn at stage n. Immediate decision is given by xn, and optimal decisions are made in accordance, tr (s , x ): Transformation function which maps the current state and • n n n decision to a new state sn+1, f ∗(s ) = f (s , x∗ ). • n n n n n

42 6.2 Application to Dynamic Decision Problems

STATE: Contribution of VALUE:

STAGE

Figure 6.1 – The basic structure for deterministic dynamic programming [Hil07]. Making a policy decision xn moves the process to a state sn+1 at stage n+1. The subsequent contribution to the objective function under an ∗ optimal policy has been previously calculated to be fn+1(sn+1). The policy decision xn also makes some contribution to the objective func- tion. Combining these two quantities provides fn(sn, xn) which is the contribution of stages n onward to the objective function. Optimiz- ∗ ∗ ∗ ing with respect to xn then gives fn(sn) = fn(sn, xn). After xn and ∗ fn(sn) are found for each possible value of sn, the solution procedure is moved back one stage. The procedure repeats itself until the initial stage is reached.

Please note that the recursion is a result of the fact that fn(sn, xn) can be ∗ written in terms of xn, sn, contribution of xn, and fn+1(sn+1) (cf. Fig. 6.1). 7. The solution procedure for this recursive relationship starts at the end and moves backward stage by stage. I.e., for each stage, an optimal policy is derived, until the initial stage is reached. The resulting (optimal) policy yields to an optimal ∗ solution for the entire problem. I.e., there is an optimal decision x1 for state s1, ∗ leading to an optimal decision x2 for state s2, concluding to an optimal decision ∗ xN for state sN .

This work deals with decision problems that are discretized in time. Consequently, the number of resulting time steps correspond to the number of stages, cf. Fig. 6.1. For a layout with only one valve prone to wear, a state sn specifies the wear level of this single valve at stage n, and the number of states at this stage n is given by the number of discretized wear levels. If more than one valve is considered, a state is represented by the combination of the wear levels of all valves. Fig. 6.2 depicts the case in which two valves and three wear levels are considered. In general, the number of states given C components and W wear levels accounts to W C . Consequently, there are 32 = 9 states per stage in the depicted example.

The decision variable xn at stage n (cf. Fig 6.1) represents the decision if valves ∗ are subject to a specific wear. Given a state s at stage n, fn(sn) corresponds to ∗ ∗ the minimum value of fn(sn, xn). Thus, fn(sn) = minxn fn(sn, xn) = fn(sn, xn). Where fn(sn, xn) = “immediate costs at stage n (costs of replace or costs if a valve is subject to wear)” + “minimum future cost (stages from n onward)”. That ∗ ∗ corresponds to fn(sn) = csxn + fn+1(sn+1), where csxn are the immediate costs at

43 6 Dynamic Programming

∗ stage n, and fn+1(sn+1) are the accumulated costs for the onward stages to the last stage.

Formally, an (optimal) policy is given by [Dom+15]:

Definition 6. A policy is a sequence of actions (xh, xh+1 . . . , xk) which transfers ∗ ∗ ∗ the system from state sh to state sk+1. An optimal policy (xh, xh+1 . . . , xk) transfers the system from state sh to state sk+1 by minimizing the objective function.

Since the stage-based objective function fn(sn, xn) (cf. Eq. 6.1) only depends on state sn and decision xn, Bellman’s principle of optimality can be formulated ∗ ∗ ∗ ∗ [Dom+15]: Let (x1, . . . , xk−1, xk, . . . , xN ) the optimal policy, which transfers the ∗ system from initial state to a final state sN . Furthermore, let sk−1 the system state at stage k 1. Then the following holds: −

∗ ∗ ∗ 1.( xk, . . . , xN ) is an optimal policy which transfers the system from state sk−1 in stage k 1 to a final state in stage N. − ∗ ∗ 2.( x1, . . . , xk−1) is an optimal policy which transfers the system from the initial state to a state s∗ in stage k 1. k−1 −

Definition 7. In this context, a Problem Pk(sk) is the task of finding the optimal policy for a state in stage k to the final stage N. The optimal objective value of ∗ this problem is designated by Fk (sk).

Definition 8. The bellman equation is given by [Dom+15]:

∗ ∗ F (s − ) = min f (s − , x ) + F (s = tr (s − , x )) x X (s − ) t−1 t 1 { t t 1 t t t t t 1 t | t ∈ t t 1 }

By using Def. 8, the following algorithm for our system can be derived:

Data: Data of the minimization problem ∗ ∗ Result: Optimal policy (x1, . . . , xN ) for t = N to 1 do solve Pt−1(st−1) for all st−1 St−1 using: ∗ ∈ ∗ Ft−1(st−1) = min ft(st−1, xt) + Ft (st = trt(st−1, xt)) xt Xt(st−1)) ∗ { ∗ | ∈ } set xt−1(st−1) = arg min ft(st−1, xt) + Ft (st) end { } for t = 1 to N do ∗ ∗ ∗ ∗ set xt = xt(st−1) and st = trt(st−1, xt ); end

Algorithm 3: Dynamic Programming using backward recursion.

44 6.3 Algorithmic Operations

0 T-2 T-1 T TIME

OPERATION SERVICE OPERATION INTERVAL INTERVAL INTERVAL

Figure 6.2 – Structure of a dynamic program representing a decision problem con- taining 3 wear levels and 2 valves. The solution procedure starts by evaluating the last time step. For a state at time T 1, all 9 possi- − ble decisions have to be evaluated (black arcs), but only the optimal decision has to be remembered (blue arc).

6.3 Algorithmic Operations

In this work, we deal with sequential decision problems that are discretized in time. We use two types of time steps (cf. Fig. 6.2 ff): During the operation interval, the valves are subject to a wear depending on the current loading case. At the service interval, a decision of whether to replace a valve has to be made. In case of two valves, the replacement can be related to one, two or none of the valves (cf. corresponding arcs at service interval in Fig. 6.2).

The solution procedure starts at the last time step. By using the concept of dy- namic programming the following holds: No matter which decisions were made before, such that a specific state at time T 1 is part of the optimal decision sequence, for an overall optimal decision sequence,− we need to chose an optimal follow state at time T . That means, knowledge of the current state and its optimal

45 6 Dynamic Programming

0 T-2 T-1 T TIME

OPERATION SERVICE OPERATION INTERVAL INTERVAL INTERVAL

Figure 6.3 – Solution procedure continues: for a state at time T 2, all 9 possible − decisions have to be evaluated (black arcs), but only the optimal deci- sion has to be remembered (blue arc). This decision incorporates the contribution of the current decision and the accumulated contributions of the subsequent decision to time step T . follow up costs conveys all the information that is necessary for determining the optimal policy henceforth.

Fig. 6.2 depicts the evaluation of one state of stage T 1. For a state of that stage, all possible decisions have to be evaluated (indicated− by black arcs). Only the optimal decision has to be remembered (represented by a blue arc). But at this stage, it is still not known, which state will be reached at time T 1. Therefore, all decisions of all states at T 1 have to be evaluated. This results− in an optimal decision x∗ for each state at stage− T 1 (cf. Fig. 6.3). − The procedure repeats itself at stage T 2, i.e. for each state at stage T 2 an optimal decision is evaluated (cf. Fig.− 6.3). The evaluation incorporates− the contribution of the immediate decision and the accumulated contributions of the subsequent decision to stage T . Consequently, the evaluation is not “greedy” in the sense of only considering the immediate decision, but takes the accumulated

46 6.3 Algorithmic Operations

0 T-2 T-1 T TIME

OPERATION SERVICE OPERATION INTERVAL INTERVAL INTERVAL

Figure 6.4 – Solution procedure has finished. For each state at time step 0, the optimal decisions, leading to stage T are known. The figures depicts the optimal policy for one state at time step 0. In this example, we start from a state in which the first valve has already reached its maximum wear level. In the next time step, the second valve thus is used. One of the valves is subsequently replaced with the costs of 1. The replaced valve can be used to satisfy the load requirements of the last time step, however it again is subject to wear. subsequent contributions into account.

Fig. 6.3 depicts the evaluation of decisions for a state in stage T 2. Both valves show a medium wear level. An optimal decision is calculated with− respect to the contribution of the immediate decision (service interval) and the accumulated con- tributions of the next state henceforth (operation interval). The optimal decision for a state is remembered (blue arc). The calculation has to be done for all states in stage T 2, since in is not known which state will be reached at that time. − The solution procedure moves backwards stage by stage. For each stage an optimal policy is derived, until the initial stage is reached. The resulting (optimal) policy yields to an optimal solution for the entire problem, i.e., we can derive an optimal

47 6 Dynamic Programming decision for each state at the beginning stage, concluding to an optimal decision to a state of the last stage. Fig. 6.4 depicts the optimal policy for an exemplary state at time step 0. Blue arcs indicate the optimal decisions for each stage. Note that for each state an optimal policy is available – leading to an optimal solution for the entire problem.

6.3.1 Benchmarks

In this chapter, the run time of the dynamic programming algorithm is investigated. The benchmarks were executed for exemplary system topologies of one, two, three or four proportional valves that were connected in parallel, since the number of system states that have to be considered is given by LN , where L is the number of wear levels and N is the number of components prone to wear. From Fig. 6.6, it can be deduced that the runtime of the algorithm depends expo- nentially on the number of discretized system states considered. On the contrary, for the application case considered, the computation of a look up table has to be executed only once, since the loading case of moving the piston in and out is con- sidered the same for each operation interval. The following recursive comparisons for the operation intervals are cheap. This is also shown by the results obtained for different numbers of operation in- tervals, i.e., coupled states. Fig. 6.5 shows that the dependence of the runtime on the number of operation intervals is negligible for the optimization problem consid- ered. Since each operation interval is coupled to a maintenance interval, a number of n operation intervals corresponds to 2n 1 stages in the dynamic programming algorithm. − Fig. 6.5 also shows an approximately quadratic dependence of the runtime on the number of wear levels considered. Overall, the dynamic programming approach is beneficial, if the number of operation intervals is high, and the number of compo- nents is manageable.

48 6.3 Algorithmic Operations

3,000 2 Service Points 5 Service Points 10 Service Points

2,000

1,000 RUNTIME in s

0 2 3 4 5 6 7 8 9 10 # LEVELS

Figure 6.5 – Runtime of the dynamic programming algorithm for the test case of four valves. The runtime increases approximately quadratic with the number of wear levels considered. The number of operation intervals has almost no influence on the runtime.

2 Levels 5 Levels 10 Levels 103 102 101 100 RUNTIME in s 10−1 1 2 3 4 # VALVES

Figure 6.6 – Run time of the dynamic pogram for 10 service intervals for different numbers of valves and different numbers of levels. An exponential growth of the runtime with the number of discretized system states can be deduced. The number of system states is given by the number of levels to the power of the number of valves.

49 6 Dynamic Programming

50 7 Dual Bounds via Lagrangean Relaxation

The basic concept of the Lagrangean relaxation is to approximate a difficult prob- lem by a simpler one: constraints that create difficulties within the solution process are removed and introduced into the objective function by penalty terms [Fis85]. A solution to the relaxed problem is an approximate solution to the original problem, and provides a dual bound, making it possible to assess the quality of primal solu- tions found by primal heuristics like dynamic programming. A good overview for the application of Lagrangean relaxation in solving integer programming problems can be found in [Fis04].

7.1 Basics of Lagrangean Relaxation

[Mar06] gives a good introduction into the basics of Lagrangean Relaxation. The following definitions are based on this reference. We begin with a mixed-integer linear program (P). Let (LP) denote problem (P) with relaxed integrality constraints, and let zLP denote the optimal value of (LP).

A1 b1 m1×n By splitting A and b such that A = , and b = with A1 R , A2 b2 ∈ m2×n m1 m2     A2 R , b1 R , b2 R , and m1 + m2 = m one can write: ∈ ∈ ∈

min cT x s.t. A x b 1 1 (7.1) A x ≤ b 2 ≤ 2 x Zn−p Rp. ∈ ×

The Lagrangean relaxation integrates the part A1x b1 of the constraints into the objective function by penalizing their violation and≤ rewarding their satisfaction. With non-negative weights λ Rm1 , λ 0, the Lagrangean function is given by ∈ ≥

T T L(λ) = min c x λ (b1 A1x) −2 − n−p p (7.2) s.t. x P = x Z R : A2x b2 . ∈ { ∈ × ≤ }

51 7 Dual Bounds via Lagrangean Relaxation

cTx

z Dual Gap

zLR

zLP

{ x | A1x b1 } { x | A2x b2 }

Figure 7.1 – The LP relaxation of a minimization problem leads to the bound zLP to the primal objective value z = cT x. The Lagrangean relaxation yields a lower bound zLR: The primal objective function is optimized on the intersection of the convex hull of the feasible set of the non- complicating constraints x A x b , and the LP relaxation of the { | 2 ≤ 2} constraints set x A x b . The bound obtained by Lagrangean re- { | 1 ≤ 1} laxation zLR is at least as tight as the bound zLP obtained by LP relaxation.

This is a relaxation of the original problem because removing the constraint A1x b1 relaxes the original feasible space, and for any fixed set of λ 0, and every feasible≤ solutionx ˜ to the original problem the following inequality holds≥

T T T T T c x˜ c x˜ λ (b1 A1x˜) min c x λ (b1 A1x) = L(λ) , (7.3) ≥ − − ≥ x∈P 2 − −
showing that the Lagrangean relaxation yields a lower bound on the objective value of the original problem 7.1. In general, for different values of λ one obtains different lower bounds. By maximizing the minimum value of the relaxed problem with regard to λ, one obtains a tighter lower bound to the original problem. This problem is called the “Lagrangean dual”:

L(λ∗) = max L(λ) λ≥0 T T (7.4) = max min c x λ (b1 A1x)) λ≥0 x∈P2 − −
According to [Mar06] the following theorems hold: Theorem 1. L(λ) is piecewise linear and concave on its domain of definition. Theorem 2. L(λ∗) = min cT x : A x b , x conv(P 2) . { 1 ≤ 1 ∈ }

52 7.1 Basics of Lagrangean Relaxation

Since x Rn : Ax b conv x Zn−p Rp : Ax b we get the following inequality{ ∈ from Theorem≤ } 2: ⊇ { ∈ × ≤ }

z L(λ∗) z , (7.5) LP ≤ ≤ MILP where zLP and zMILP are the optimal objective values of the LP relaxation and the original mixed-integer linear problem, respectively. This inequality can also be motivated graphically, cf. Fig. 7.1. The optimization of the Lagrangean dual can be interpreted as optimizing the primal objective function on the intersection of the convex hull of the feasible set of the non-complicating constraints x A2x b2 and the LP relaxation of the constraints set x A x b [Fra05]. { | ≤ } { | 1 ≤ 1}

7.1.1 Subgradient Method

From Eqs. 7.4 and 7.5 follows that the best choice for λ corresponds to the opti- mal solution to the dual problem. We have seen, that the Lagrangean function is concave and piecewise linear, cf. Theorem 1. A classical approach of maximizing a concave function would be to look at the corresponding minimization problem and employ gradient descent, where, starting from an initial point, iterative minimiza- tion is achieved by taking steps in the negative direction of the function gradient. This method is however not applicable since the Lagrangean function is non-smooth and therefore not differentiable everywhere. Therefore, we adapt the method at points where the function is non-differentiable. Definition 9. Let L :Rm1 R be a concave function. The vector g Rm1 is a subgradient of L at λ(0) if → ∈

L(λ) L(λ(0)) gT (λ λ(0)) − ≤ − m for all λ R1 ∈ (0) m1 (0) (0) (0) Theorem 3. Given λ R+ and an optimal solution x . Then g = A1x ∈ (0) − b1 is a subgradient of L in λ . λ∗ maximizes L 0 is a subgradient of L in λ∗. ⇔ The subgradient method starts with λ(0) and computes

x(0) = argmin cT x (λ(0))T (b A x): x P 2 . (7.6) { − 1 − 1 ∈ } λ∗ is found iteratively: Starting from an initial value λ(0) we use the simple iteration formula λ(k+1) = λ(k) + µ(k)g(k), (7.7) where λk is the k-th iterate, µ(k) is the step size in iteration k and g(k) any subgra- dient of L at λ(k).

53 7 Dual Bounds via Lagrangean Relaxation 7.2 Lagrangean Decomposition

Lagrangean Decomposition is a decomposition method based on Lagrangean relax- ation. In general, decomposition methods solve a problem by breaking it up into smaller ones, which can then be solved separately, either in parallel or sequentially [Boy+07]

Some problems can be easily decomposed. If the vector of decision variables can be partitioned into subvectors x1, , xk, and each constraint involves only variables from one of these subvectors, the··· problem is called block separable. In this case each subproblem corresponding to one block can be solved independently, and the problem is trivially parallelizable.

Usually, there is however some coupling between the subvectors, like in the problem structure investigated in this thesis, c.f. Fig. 3.2. Despite being not trivially par- allelizable, many of these problems can still be tackled by decomposition methods, one of which is Lagrangean Decomposition.

Given a MILP with the special structure as shown in Fig. 3.2:

min cT x s.t. Ax b (7.8) D x≤ e , i = 1, , n i ≤ i ··· x Zn−p Rp. ∈ ×

In this case, the only thing preventing the problem from being block separable are the complicating constraints Ax b. Lagrangean Decomposition is a special case of Lagrangean relaxation that makes≤ use of this special problem structure. The first step in decomposing is to introduce new local variables yi for each set of constraints Di and adding constraints to equate these new variables:

min cT x s.t. Ax b ≤ Diyi ei, i = 1, ,N (7.9) y = x≤, i = 1, ··· ,N i i ··· x, y Zn−p Rp. ∈ × The newly added equality constraints make sure, that the new local versions of the variables in the different subproblems are consistent to each other. However, by now, old complicating constraints have just been replaced by new ones. Only by dualizing these consistency constraints and adding them to the objective function, we get a separable problem. The dual problem obtained by Lagrangean decompo- sition is

54 7.2 Lagrangean Decomposition

cTx

z

zLD

zLR

zLP

{ x | A1x b1 } { x | A2x b2 }

Figure 7.2 – Graphical interpretation of the Lagrangean decomposition. The pri- mal objective function is optimized on the intersection of the convex hull of constraints sets x A x b and x A x b and yields a { | 2 ≤ 2} { | 1 ≤ 1} dual bound zLD greater or equal to the one found by Lagrangean re- laxation, zLR, and LP relaxation, zLP.

T N min c x + i=1 λi(yi xi) s.t. Ax b − (7.10) D y≤ Pe , i = 1, ,N i i ≤ i ··· x, y Zn−p Rp. ∈ × This new problem can now be decomposed and solved independently for each block Di, i.e., for each time step or operation interval independently. Lagrangean decomposition is a special case of Lagrangean relaxation. Every con- straint in the original problem appears in one of the subproblems. Fig. 7.2 shows the graphical interpretation of Lagrangean Decomposition: We optimize the primal objective function on the intersection of the convex hulls of the constraint sets. Like Lagrangean relaxation, Lagrangean decomposition yields a lower bound to the original problem. Guignard and Kim [GK87] showed, that the bound provided by Lagrangean decomposition, zLD, is at least as tight as the one provided by Lagrangean relaxation, zLR:

z z z z (7.11) MILP ≥ LD ≥ LR ≥ LP The next sections will show how Lagrangean decomposition can be applied to the application example considered in this work, a hydrostatic transmission system with components prone to the dynamic process of wear.

55 7 Dual Bounds via Lagrangean Relaxation 7.3 Wear-Induced Component Replacement

In this section, we will show how dual decomposition can be applied to the op- timization of a technical system whose components are subject to the dynamic process of wear. Whenever a component reaches a maximum wear level, it has to be replaced. The mathematical optimization model presented in Section 4.1 allows to anticipate the resulting maintenance costs as early as during the design process. However, the computing time for solving the corresponding MILP with commer- cial solvers rises sharply with the number of operating intervals considered. Dual decomposition allows to decouple successive operating intervals and to calculate good dual bounds for the replacement costs.

7.3.1 Dual Decomposition Approach

For a given purchase decision and without the consideration of wear-induced re- placements, the optimization problem shown in Section 4.1 is trivially paralleliz- able since each operating interval s = 1,...,T in between maintenance points corresponds to one subproblem and there is no coupling between the intervals. If one wants to balance initial investment and maintenance costs during the expected operating time of the system, several complicating equations have to be added to the basic model.

When operating the system, the wear of a valve produced in one operating interval leads to an already worn valve at the beginning of the next interval, if this valve is not replaced. This couples the subproblems for the different operation intervals and makes up the complicating constraints.

First we introduce local copies of these variables for each operating interval and replace Eq. 4.17 by:

(v , v ) V, s = 1, ,T : wend wbegin + ∆w , ∀ i j ∈ ∀ ··· s,vi,vj ≤ s,vi,vj s,vi,vj (v , v ) V, s = 1, ,T : wend wbegin + ∆w w r , ∀ i j ∈ ∀ ··· s,vi,vj ≥ s,vi,vj s,vi,vj − max · s,vi,vj (v , v ) V, s = 2, ,T : wbegin = wend . ∀ i j ∈ ∀ ··· s,vi,vj s−1,vi,vj (7.12)

To decouple the constraints of the different operation intervals, we introduce a variable ys,vi,vj for each coupling constraint that represents the common value of both coupled variables:

56 7.3 Dual Decomposition

(v , v ) V, s = 1, ,T : wend wbegin + ∆w , ∀ i j ∈ ∀ ··· s,vi,vj ≤ s,vi,vj s,vi,vj (v , v ) V, s = 1, ,T : wend wbegin + ∆w w r , ∀ i j ∈ ∀ ··· s,vi,vj ≥ s,vi,vj s,vi,vj − max · s,vi,vj (v , v ) V, s = 2, ,T : wbegin = wend = y . ∀ i j ∈ ∀ ··· s,vi,vj s−1,vi,vj s,vi,vj (7.13)

According to [Boy+07], we form the partial Lagrangian of this problem by adding the complicating constraints to the objective function for each valve (vi, vj) V and each operating interval s = 1,...S: ∈

begin begin end end z + λ w y − + λ w y s,vi,vj s,vi,vj − s 1,vi,vj s,vi,vj s,vi,vj − s,vi,vj 6 6 (viX,vj )∈V sX:s=1   (viX,vj )∈V sX:s=S   (7.14)

To find the dual function, we first minimize over the variables ys,vi,vj , which results in the conditions

λbegin = λend , s = 1,...S 1, (v , v ) V (7.15) s+1,vi,vj − s,vi,vj ∀ − ∀ i j ∈ for boundedness. This condition states, that for each coupling constraint, the sum of the Lagrangean multipliers must add to zero. The dual of the original problem is:

begin begin end end max z + λs,vi,vj ws,vi,vj + λs,vi,vj ws,vi,vj ) 6 6 (viX,vj )∈V sX:s=1 (viX,vj )∈V sX:s=S (7.16) s. t. λbegin = λend , s = 1, . . . S, (v , v ) V s,vi,vj − s,vi,vj ∀ ∀ i j ∈

Now, the single subproblems can be solved separately. Dual decomposition with a projected subgradient method for the master problem results in the following algorithm: [Boy+07]

57 7 Dual Bounds via Lagrangean Relaxation

begin

Intialize price vector λ so that it satisfies Eq. 7.15, e.g. λ = 0; while not stopcriterion do

Optimize subproblems separately;

begin end Obtain ws+1,vi,vj , ws,vi,vj ; begin end Compute average value of ws+1,vi,vj and ws,vi,vj begin end yˆs,vi,vj := 0.5 ws+1,vi,vj + ws,vi,vj ;  begin end Update prices on ws,vi,vj and ws,vi,vj λbegin := λbegin + µ (ˆy wbegin ); s,vi,vj s,vi,vj k s,vi,vj − s,vi,vj λend := λend + µ (ˆy wend ); s,vi,vj s,vi,vj k s,vi,vj − s,vi,vj end

end

Algorithm 4: – Lagrangean decomposition algorithm with a subgradient method for master problem.

7.3.2 Verification For Small Test Cases

The subgradient method was tested with different step sizes µ. Different choices of step size and target values for the method have been investigated in the literature [Gof77]. The fundamental theoretical result [Pol67] is that L(λ) will converge to its maximum L∗ under the conditions

lim µ(k) = 0, (7.17) k→∞ and ∞ µ(k) = , (7.18) ∞ Xk=0 where µ(k) is the step size in iteration k. The step size most commonly used in practice [Fis04] is (k) (k) c zub zdual(λ ) µ(k) = − , (7.19) Ax(k) b 2 k − k

58 7.3 Dual Decomposition

(k) (k) where c is a scalar satisfying 0 < c 2, and zub is an upper bound on zdual that can e.g. be obtained by applying a heuristic≤ to the original MILP. This step size showed the best results and was chosen in this work, with an upper bound zub that was deduced as follows:

z = (T 1) V , (7.20) ub − · | | where T is the number of maintenance points and V the number of available valves. In the following section, the evolution of the dual| | bound will be shown for several test cases.

One Load Case, One Valve

If one wants to verify the convergence of the dual decomposition algorithm for the small test case of only one component, i.e., one valve, the problem has to be sim- plified since only one load case (moving the piston out or in) can be satisfied with this topology option. For the following small two small test cases we investigate the load case of the piston running out, cf. Fig. 4.1(b).

Fig. 7.3 depicts schematically the solution of the corresponding MILP for this sim- ple test case: If we invest in only one valve, this valve has reached a wear level of almost wmax at the end of the second operating interval and is subsequently replaced. With the replaced valve, the next operating interval can be performed and no additional replacement is necessary. Therefore, the objective function, that ∗ counts all replaced components, accounts to zMILP = 1. In comparison, Fig. 7.4 depicts the evolution of the dual bound obtained by dual decomposition over the course of 500 iterations. The best lower dual bound found accounts to approx- imately z 0.3482. This bound is greater than the bound obtained by LP LR ≈ relaxation which accounts to zLP = 0. Since the objective of the original problem ∗ can only take integer values, we can conclude that zMILP 0.3482 = 1 which exactly corresponds to the solution found when solving the≥ original d MILP.e

The next test cases also feature the simple topology of one valve and one load case, but we consider more operating intervals. Fig. 7.4 shows that for these simple test cases the Lagrangean relaxation based dual decomposition yields good lower bounds. E.g., for the test case of eleven operating intervals, cf. Fig. 7.4(d), we find a dual bound of zLR 4.1664. Again, since the objective of the original ≈ ∗ problem can only take integer values, we can conclude that zMILP 4.1664 = 5 ∗ ≥ d e which again corresponds to the solution zMILP = 5 found when solving the original MILP.

59 Modellierung von Wartungsintervallen

푉 푤−1,푙,푒 ,푒 ≤ 푣푚푎푥 + δ ∗ 퐷푤,푙,푒 ,푒 7 Dual Bounds via Lagrangean푖 푗 Relaxation 푖 푗

풘max

횫풘ퟐ

풘ퟑ WEAR

s=1 s=2 s=3 TIME

2. Juli 2015 | Prof. Dr.-Ing- Peter Pelz| Auslegung eines hydrostatischen Getriebes | Lucas Farnetane 1 Figure 7.3 – Evolution of wear, and replacement strategy for a simple test case with one valve. After two operating intervals, the valve has to be replaced. The objective accounts to one.

One Load Case, Two Valves

To further verify the bounds obtained by dual decomposition we consider another test set. In this next small test case, two valves are connected in parallel. Again, for simplification we only consider the load case for the piston running out. Both valves are used until they both have reached the maximum wear level and have to be replaced. The results obtained from dual decomposition are shown in Fig. 7.5.

7.3.3 Benchmarks

The approach of decoupling subsequent time steps was verified for a larger test case consisting of four valves. Considered were two different scenarios: In one scenario, 100,000 load cycles had to be executed in between the service points, in the other scenario 150,000 load cycles. The dual decomposition is compared to the primal and dual bounds obtained by the commercial solver CPLEX. Table 7.1 gives an overview of the benchmarks that were executed for the load case with 100,000 cycles per operation interval. For up to four operation intervals, CPLEX is able to find the global optimal solution in less than 24 hours. For six, seven and eight intervals cplex cannot find the global optimal solution. In these cases, the best primal and dual solutions, zprimal and zdual, obtained so far are given. The lower bound zLD surpasses the bound found by LP relaxation in each case . For a number of seven operation intervals, the dual decomposition approach is able to find better bounds than the one obtained by CPLEX. Table 7.2 shows the corresponding results for the load case with 150,000 cycles per operation interval. In this case, more valves have to be replaced. Again, the dual decomposition yields higher bounds the the LP relaxation. Starting from a number

60 7.3 Dual Decomposition

intervals iterations runtime in s zLD zprimal zdual zLP 4 500 3702 0.3199 1.0 1.0000 0.0000 4 1000 11515 0.3514 1.0 1.0000 0.0000 4 2000 27555 0.3514 1.0 1.0000 0.0000 5 500 4781 1.1034 2.0 2.0000 0.8329 5 1000 15098 1.1535 2.0 2.0000 0.8329 5 2000 36306 1.2017 2.0 2.0000 0.8329 6 500 5537 1.9178 4.0 2.0000 0.8795 6 1000 17371 1.9178 4.0 2.0000 0.8795 6 2000 42797 1.9278 4.0 2.0000 0.8795 7 500 6533 2.7464 6.0 2.0173 2.0173 7 1000 20831 2.7464 6.0 2.0173 2.0173 7 2000 49967 2 .8576 6.0 2.0173 2.0173 10 500 8705 4.8567 10.0 7.6796 2.6580 10 2000 79279 4.9902 10.0 7.6796 2.6580

Table 7.1 – Dual bounds obtained by Lagrangean relaxation for the case of 100,000 load cycles per operation interval. of six operation intervals, the bounds obtained by dual decomposition are better than the ones obtained by CPLEX.

61 7 Dual Bounds via Lagrangean Relaxation

1 2 0 1 0 1 − 1 2 − − 2 3 − − 3 DUAL BOUND DUAL BOUND − 4 4 − − 120 140 160 180 200 200 300 400 500 ITERATIONS ITERATIONS (a) Three operation intervals. (b) Five operation intervals.

3 4 2

2 1 DUAL BOUND DUAL BOUND 0 0 200 300 400 500 200 600 1,000 1,400 ITERATIONS ITERATIONS (c) Seven operation intervals. (d) Eleven operation intervals.

Figure 7.4 – Evolution of the dual bound with the number of iterations for different test cases. Each test case has a different amount of operation intervals. The objective of the corresponding MILP is shown by the dashed blue line. In each case, Lagrangean based dual decomposition yields a bound greater than the one obtained by LP relaxation (dashed red line).

62 7.3 Dual Decomposition

1 0 0 1 − 2 2 − − DUAL BOUND 4 DUAL BOUND − 350 400 450 500 250 300 350 400 450 500 ITERATIONS ITERATIONS (a) Three operation intervals. (b) Five operation intervals.

2 4 3 1 2 0 1 1 0 − 1 − DUAL BOUND DUAL BOUND

300 350 400 450 500 500 1,000 1,500 2,000 ITERATIONS ITERATIONS (c) Seven operation intervals. (d) Eleven operation intervals.

Figure 7.5 – Evolution of the dual bound for two valves. Each test case has a different amount of operation intervals. The objective of the corre- sponding MILP is shown by the upper dashed blue line. In each case, Lagrangean based dual decomposition yields a bound greater than the one obtained by LP relaxation (lower dashed red line).

63 7 Dual Bounds via Lagrangean Relaxation

operation intervals iterations runtime zLD zPRIMAL zDUAL zLP 4 500 4906 2.2042 4.0 3.0000 0.7948 4 1000 12489 2.2221 4.0 3.0000 0.7948 4 2000 28044 2.2527 4.0 3.0000 0.7948 5 500 6103 3.4810 6.0 5.0000 1.4935 5 1000 16260 3.5368 6.0 5.0000 1.4935 5 2000 35023 3.5368 6.0 5.0000 1.4935 6 500 7411 4.7707 8.0 4.6104 2.1922 6 1000 90931 4.8074 8.0 4.6104 2.1922 6 2000 41702 4.8076 8.0 4.6104 2.1922 8 500 7245 7.3844 12.0 6.3124 3.5896 8 1000 24566 7.4190 12.0 6.3124 3.5896 8 2000 54179 7.4190 12.0 6.3124 3.5896 10 500 8720 9.9448 17.0 8.0000 4.9870 10 2000 115171 9.9749 17.0 8.0000 4.9870

Table 7.2 – Dual bounds obtained by Lagrangean Relaxation for the case of 150,000 load cycles per operation interval.

64 8 Conclusion and Outlook

This work was written as part of the Collaborative Research Centre 805, whose aim is to control uncertainty in load-carrying structures in mechanical engineering. In this context, the question of finding an optimal structure for a technical system was investigated for the case of a truss topology demonstrator. This dissertation examines the issue of structural uncertainty for a hydraulic system. In this work, the TOR Ansatz for system design was applied to find an optimal sys- tem structure, operation and maintenance strategy for a hydrostatic transmission system prone to wear. By this holistic approach a larger potential for optimization opens up, since – instead of optimizing single components – the system structure itself is questioned. However, the increased potential is accompanied by an in- creased level of complexity. By the combination of different components, a variety of possible structures arises, resulting in a combinatorial explosion of the solution space. Especially system synthesis problems in which dynamic processes like wear have to be modeled can often not be solved to global optimality in reasonable time by means of standard discrete optimization techniques. In this work, the specific problem structure of such problems was exploited to derive primal solutions and dual bounds by decomposition methods. The optimization problem was modeled as a mixed-integer linear program which is based on the graph representation of the system synthesis problem. In case of a series-parallel network, all possible system graphs can be represented by trees without internal nodes of degree 2 (series-reduced trees). The approach has been employed in two master’s thesis in the context of TOR [Bau12; Utz14]. This work shows how the approach can be transferred to the the more general case of a non-series parallel network constituting a hydrostatic transmission system. Using problem specific knowledge, the graph is divided into series-parallel subgraphs that are combined to a valid topology option. In this way the tree representation can still be exploited to elegantly traverse the solution space of discrete structure decisions using a local search. The developed primal heuristic based on the combination of simulated annealing and dynamic programming is able to find solutions to the optimization problem in cases in which a commercial solver cannot find any feasible solution in reasonable time.

65 8 Conclusion and Outlook

The challenge arising from the temporal coupling of subsequent operating intervals due to dynamic wear processes was tackled in two ways: Taking advantage of the optimality principle of Bellman an optimal replacement strategy in case of wear induced component failure is calculated by backward re- cursion. This makes it possible to anticipate the effect of long term replacement costs due to wear processes as early as during the design phase. Via a dual decomposition approach individual operating intervals were decoupled. This allowed to derive dual bounds and to estimate the costs of spare parts. The approach looks promising for the case of more operating intervals. In this case bounds found by cplex were outperformed. In general, a decreasing quality of the dual bound with the number of relaxed constraints is expected. Since the approach showed good results for the decoupling of operating intervals, integrating the investment decision into the dual decomposition could still be promising. By means of the models and the algorithms developed, wear-related maintenance costs and downtime of a possible system layout can be anticipated in the early phase of structure determination. Thus, this thesis lays a foundation for the control of structural uncertainty in real systems.

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