A study of train delay propagation using max-plus algebra and extensions of the max-plus model.

In cases that the voltage source is made of multiple cos’s and we can compensate the current, it is always possible with the first Cauer form. This is caused by the shape of the found transfer function Z(s). This transfer function is a fraction where the odd powers of s are in the numerator, while in the denominator are only even powers of s. In this way the conditions 1 and 3 in chapter 4.2 are satisfied. For this special type of Z(s) satisfying condition 2, is having positive coefficients in the First Cauer form, hence the conclusion. In the tables in this thesis, it seems that when the first Cauer form works, the other forms areworking as well. It seems that the method used in this thesis to find a controller only works for small R, L and C. In the example the improved PF is optimal, like one would expect. I conclude that the method used in this thesis is working fine.

This paper is a research in positive controllability and stabilizability for discrete-time linear systems. Several conditions are given and explained for both single input as multiple input systems. Also the connection between controllability and reachability will be mentioned. Furthermore, a method for multiple input positive controllability is analysed.

In this thesis max-plus algebra is introduced and applied to the problem of controlling train delays. Two control strategies for the propagation of delays are discussed. The first is by letting certain trains run faster when a delay is detected, the second is by breaking connections between trains that have to wait for each other in order to enable passengers to changeover from one train to the other. The goal is to understand how to model the propagation of delays when different control strategies are applied, in order to provide train operators tools for making quick decisions on how to intervene when a delay is detected.
The models provided in the report are in the form of max-plus-linear systems and switching max-plus linear systems. These can be programmed in Python to automate the decision making. The report starts with providing a basic understanding of max-plus algebra, where also max-plus linear systems and switching max-plus linear systems are explained. Subsequently, a railway network is designed that serves as an example during this thesis. This railway network is modelled into a max-plus linear system and, additionally, a desirable train timetable for passengers is designed for this network by means of the power algorithm. The main results are two switching max-plus linear systems that model the propagation of delays when the two control strategies are applied. The report ends with a larger railway network at which all acquired knowledge is applied. It can be concluded that the models in this thesis provide methods to calculate exactly how the delay propagates through the network when certain control strategies are applied and, based on that, decisions can be made quicker. Moreover, it is possible to calculate the consecutive departure times. As a result the passengers can be informed quickly about the new departure times as a consequence of the delay and how long it will take for the trains to run according to timetable again. This thesis adds the modelling of faster running trains to existing literature. We have seen that speeding up trains is also a control strategy to solve delays and can be modelled systematically.

The aim of this Bachelor Thesis is to extend the orthogonal projection method described in 'An Orthogonal Projection Method for Computing Active, Reactive and Scattered Power and its Application to Compensator Design' by Jacob van der Woude and Dimitri Jeltsema. In the aforementioned article, a method is described to calculate the active, reactive and scattered power of a RLC network. It uses an orthogonal projection of the current on the space of (anti)derivatives of the voltage. This method can be used to improve the power factor of the network, as it also shows how to design a compensator for the reactive power. The compensator consists of a parallel connection of an electromagnetic coil and a capacitor. Two working examples are given. This research shows that Van der Woude and Jeltsema's method is not always applicable. It often results in negative values for the coil and the capacitor, although this has no physical meaning. When the given voltage consists of more than two frequencies, the method is also inapplicable. This research gives a condition that ensures an applicable execution of this method. It also shows when these conditions are met. We show how the method can be extended to a working method for any RLC network, provided the voltage consists of two frequencies. It uses a second Foster canonical form to expand the initially proposed compensator. Depending on the coefficients given by the orthogonal projection, a different type of compensator is needed. It is also described how a fitting compensator can be chosen, and how the coefficients for the coils and capacitors can be determined.

In this research, known theory and methods for electric networks are translated and applied to gas transport networks.

The networks are represented by directed graphs with their corresponding incidence matrices. The theory about these is extensively discussed in addition to their relevance to Kirchoff's Circuit Laws.

Several components in a gas transport network have their behaviour compared to their counterparts in electric networks. These are the pipe segments, valves and compressors. More importantly, their corresponding equations for the relation between pressure loss and volumetric flow are given.

The known methods for reducing series and parallel connections in electric networks are translated to the case of a gas pipe network. This can be used to reduce complicated networks containing pipe segments in a series or parallel connections to more manageable networks.
Additionally, the Δ/Y transforms are briefly discussed.

Lastly, a theorem guaranteeing the uniqueness of network variables is discussed. This is applied by using numerical methods, in particular the multivariate Newton-Raphson method. An algorithm is created that returns all unknown network variables after being given a specific set of known variables.
It is first constructed to work only for graphs in which each arc represents a pipe segment. Afterwards, the algorithm will be expanded to also allow check valves and compressors and it is explained how further component types can be added.
The algorithm is eventually applied to an example of a large network based on a schematic approximation of the Dutch gas transport network.

A switching max-plus linear model is a framework to describe the discrete dynamics of the timing of events. To influence these systems one can choose the routes of jobs and the orderings of operations as input for the system. In this thesis the techniques of model predictive control are used to find good input values. The problem of finding the optimal input is however NP-hard, which means there is no guarantee to find the optimal in a reasonable amount of time. This is an issue for model predictive control on real applications of the switching max-plus model. In applications, on-line performance is used where there is limited time to compute the input values for control.\\ This thesis takes a look into methods to reduce the computational complexity of the MPC-SMPL problem. Alternative formulations such as reparameterization, model based-partitioning and the cutting plane method are developed and tested for the MPC-SMPL problem. To solve the MPC-SMPL problem 3 heuristics are designed and implemented for simulation. The heuristics are partition-based optimization, tabu search and simulated annealing. The goal is to find a strategy that obtains the best solution to the problem in a limited amount of time.

In dit onderzoek hebben we een nieuwe dienstregeling voor tramlijn 19 opgesteld. Deze tramlijn rijdt nu tussen Delft en Leidschenhage. In 2020 zal het traject worden uitgebreid met haltes op de campus van de TU Delft. In dit onderzoek hebben we twee dienstregelingen opgesteld: één voor in de spitsuren, en één voor in de daluren. Er zijn een aantal voorwaarden meegenomen in het onderzoek. Zo moet de dienstregeling aansluiten op de collegetijden van de studenten die gebruik maken van tram 19. Ook moeten we de voorwaarden die we in de huidige dienstregeling vinden, meenemen in het nieuwe traject. Denk bijvoorbeeld aan de frequentie waarin trams rijden, of wanneer de dienstregeling start op een dag. Daarnaast hebben we onderzocht hoe het beste kan worden overgegaan van een spitsuur dienstregeling naar een met daluren. Dit alles is gedaan met behulp van max-plus algebra, waarbij het nemen van een maximum en optellen, de conventionele operaties van optellen en vermenigvuldigen vervangen. Het power algoritme helpt ons bij het kiezen van de juiste vertrektijden, zodat we een constant vertrekschema krijgen. Ook maken we gebruik van Petrinetten omhet gehele netwerk temodelleren. In dit onderzoek wordt steeds eerst gekeken naar een kleiner of versimpeld tramnetwerk. Hierna passen we de theorie toe op het toekomstige traject. Op het einde hebben we een dienstregeling kunnen opstellen voor tramlijn 19. In deze dienstregeling vindt afwisseling plaats tussen spits-en daluren, met elk een andere dienstregeling en hoeveelheid trams. Hierbij is rekening gehouden met de voorwaarden die wij voor ons model hebben opgesteld.

There is an increasing amount of attention for the application of model predictive control (MPC) for the optimal control of multi-purpose multi-reservoir systems (Tian et al. (2015), Raso (2013)). A major advantage of MPC is that multi-variate system dynamics and constraints of the control problem can be explicitly incorporated in the method. Challenges for application to reservoir systems include nonlinearities in the system dynamics and objectives and the large-scale optimization problem that arise when dealing with large systems (Lin and Rutten, 2016).

In this research a two-stage MPC method is presented for the optimal operation of a system of 21 reservoirs for hydropower, irrigation and flood management. The two-stage approach divides the problem into subproblems using the structure of the dynamical system (parallel connection of the reservoirs to the river) and periodicity of the external disturbances to the system (wet season and dry season). This approach reduces computational time for solution of the problem by using a decentralized approach (Christofides et al., 2013) for periods without flood risk, allowing parallel optimization of the reservoirs. In periods with flood risk, a coordinated MPC approach (Rawlings and Stewart, 2007) reduces the size of the problem while optimizing for the system-wide objective to mitigate flood. Spillway dynamics were defined by a complementarity constraint and included in the objective function using the penalty method (Pecci et al., 2017). Results showed appropriate behavior of the flows through the spillways. The main challenges for solving the nonlinear optimization problem were the scaling of the problem, the normalization of terms in the objective function, determination of appropriate penalties for the spillway constraint and the increasing size of the problem for longer optimization horizons.

The method was applied to a system in the Sittaung river basin, which suffers from frequent floodings (Rest, 2015). Results showed that trade-offs between irrigation, hydropower and flood mitigation for this basin are limited. The main limitation for flood mitigation by the reservoir system are the conduit capacities and optimization horizon of the MPC system. Results for the case study were promising and indicate clear directions for future research and necessary steps to be taken for practical implementation of a MPC system in the Sittaung river basin.

Gasunie Transport Services (GTS) is the operator of the natural gas transmission network in the Netherlands. This network is an entry-exit system where entry and exit capacities can be booked by parties called shippers. Variations in usage lead to an enormous number of possible entry-exit combinations. GTS must be able to accommodate all realistically possible gas transport scenarios that result from these entry-exit combinations. In principle, hydraulic testing of all these scenarios is required to see if the current network is optimally sized for this task. However, calculating all scenarios is extremely time consuming, so a smaller set of severe scenarios that also covers the less severe ones, is needed to represent the complete set.

Reduction of the complete set of transport scenarios is a mathematically challenging task. The current method to reduce the number of these scenarios is workable and probably meets the requirements of day-to-day planning calculations at GTS. This method makes use of the so-called end point representation, which describes transport scenarios by their capacities on entry and exit points only, while e.g. the flow pattern is unknown. The distance function for the current reduction method measures the difference between scenarios by comparing the capacities on end points, where the end points are correlated by their respective transport distances.

A new representation is introduced in this report: the flow representation. This representation makes use of the flow patterns that emerge from balanced combinations of entry and exit capacities. A flow pattern follows from an entry-exit combination by determining its minimum associated transport load. Compared to the end point representation, the flow representation is more intricate to obtain (more calculations are required to get flow patterns), but the result is a more accurate representation in terms of the transport physics of the network. The pay-off is that the distance function for the flow representation can be a lot simpler, e.g. a weighted norm which approximates the transport effort. It also turns out to be more adaptable. For example the diameter, pressure drop and other network information can easily be included in this weighted norm.

In this report both representations are compared. However, from the experiments no final conclusion can be given for which representation has a better performance. Future studies, e.g. involving hydraulic calculations, are recommended to conclude this matter.

Dit project behandelt de programmering en toepassing van een taxiservice, waarbij de passagiers zich vanaf of naar het treinstation willen verplaatsen. Met behulp van geheeltallig lineair programmeren wordt bepaald welke ritten door de taxi's moeten worden gereden, om de winst te maximaliseren. Dit model is gebaseerd op het model wat beschreven is in het verslag van Xiao Liang et al. uit 2016.
Er worden twee modellen vergeleken: in Model 1 is het systeem vrij om verzoeken te accepteren of te weigeren, terwijl in Model 2 per zone beslist wordt of alle ritten al dan niet worden. De taxiservice wordt eerst toegepast op kleine schaal, waarna enkele aanpassingen gedaan worden om het odel ook op grote schaal te kunnen toepassen. Bij de toepassing op kleine schaal wordt altijd verlies gemaakt, omdat de ratio taxi's per zone erg hoog is. Voor de toepassing op grote schaal blijkt dat het model voor veel taxi's sterk overeenkomt met het model van Liang, maar voor kleinere hoeveelheden taxi's minder omdat het genereren van ritten in Liang's model minder homogeen gedaan wordt. Het optimale aantal taxi's om te gebruiken is altijd 20 of 40.

In 1969, the theorem of Reichert was formulated. Since then, two proofs have been found, both by Z. Jiang and Malcolm C. Smith, one of which in collaboration with Sara Y. Zhang. These proofs are based on the transfer functions of the electrical networks.
The aim of this paper is to work toward a proof based on graph theory, linear algebra and differential equations. Whereas a proof has not been found, this paper provides insight on what the theorem of Reichert means, on ways to describe graphs and electrical networks with matrices and on the connection between those matrices. Lastly, we will take a look at Kron reduction: a way to remove resistors from resistive networks by manipulating their descriptive matrices.

Een BacheloreindProject over het verbeteren van de powerfactor in elektrische netwerken. Door een groter aantal frequenties in de spanning óf door bepaalde componenten in de belasting is er een compenserend gedeelte in het netwerk nodig en / of mogelijk. Met welke schakelingen, componenten en karakteristieke waarden dit gedaan kan worden, zal per situatie verschillen.

Dit bacheloreindverslag gaat over Max-plus Algebra. Dit is een algebraïsche structuur die gebruikt kan worden om roosterplanning te modelleren. In plaats van de normale optelling en vermenigvuldiging worden de operaties 'maximum nemen' en optellen gebruikt. Wanneer Max-plus Algebra wordt gebruikt in matrices, spelen de eigenwaarden en eigenvectoren van deze matrices een rol. Dit alles wordt toegepast in een voorbeeld in de luchtvaart, waar de max-plus algebra wordt gebruikt om een optimale dienstregeling te vinden in een netwerk met twee hubs op verschillende contintenten.