Sensitivity analysis based on single delays

It is known that in a railway network the change of the departure or arrival time of one train can a have small effect, but the change of the parameters of other trains can lead to an immense effect on the whole system. In the following it is shown that the proposed model structure is capable of showing the most delay-sensitive parts of the network.

To investigate the effect of a given train’s delay on the network, the following setup was used. Two different cases were examined introducing 5 and 10 minutes of individual initial delay to a given train, respectively. Note that in case of this special setup the introduced total

Figure 4.8: Dependence between the introduced total initial delays and the solution times in case of dependent and independent model formulation. The power regression lines has the following parameters and correlation coefficients. In case of the dependent model: 5·10⁻⁹x^3.18, R² = 0.831, in case of the independent model: 4.155·10⁻⁶x^2.43,R² = 0.809.

Figure 4.9: Comparison of the solution times of the dependent and independent model formulation in case of several different delay scenarios. Scenarios are plotted in an increasing order w.r.t. the solution time in case of dependent models. As it is shown in Fig. 4.8, this has a strong correlation with the total initial delay introduced into the given scenario.

Figure 4.10: Result of delay-sensitivity tests in case of uncontrolled mode. Both the scenarios having 5 and 10 minutes initial delay are presented.

initial delay is equal to the individual delay of the given train. The whole network model with N_p= 2 was generated and simulated over the prediction horizon. Both uncontrolled case and controlled case were evaluated. This procedure was iterated over all trains in the network.

This setup enables us to analyze the effect of delay on a given train. If the selected train is a bottle-neck node in the network, then the introduced initial delays will end up in high secondary delay values, showing that many other trains are effected by the delay of this specific train.

The generated results can be seen in Fig. 4.10 and in Fig. 4.11. In case of the 5mininitial delay, the average of secondary delay in uncontrolled mode is 778.37min (s= 11.95min) while in controlled mode 413.96min(s= 10.37min) is achieved. In case of 10mininitial delay, the average of the secondary delays is 800.99 min(s= 39.65min) and 430.49min (s= 27.08min) in uncontrolled and controlled case, respectively. The results are concurring with the expectations, namely that the effect of a relatively small delay (5 min) can be handled more effectively than the effect of a larger delay (10 min) because of the larger scale of delay propagation. These results also confirm that the proposed control technique can effectively reduce the amount of the delay in the network.

It should be noted that this algorithm works with the train runs in the model. As it is introduced in Subsection 4.1.1 a train run is a representation of a specific train over a given track. Handling train runs brings the advantage that besides being able to analyze the delay-sensitivity of a train we can also determine the most delay-sensitive track in the train’s route.

4.4 Summary

In this Chapter a railway scheduling problem is formulated as an MILP and solved in an MPC architecture. The model is based on the one presented in [21]. The aim of the control method is to minimize the total delay of the trains over the prediction horizon. Due to the

Figure 4.11: Result of delay-sensitivity tests in case of controlled mode. Both the scenarios having 5 and 10 minutes initial delay are represented. Delay values are clearly smaller than in Fig. 4.10.

computational complexity of the emerging MILP problem and the need of an algorithm which can solve the rescheduling in a reasonable time limit, it was necessary to speed up the solution with proper handling of the original problem. Hence, the obtained constraint set has been reformulated on two different ways, resulting in different model structures. The time consumption of the solutions have been compared to each other in case of both model forms, having different delay scenarios. The performance of the proposed control technique to minimize total secondary delays has been analyzed.

The main contributions presented in this Chapter and summarized in Thesis II. are the following: using a track-based ordering a significant speedup can be achieved during the solution process. By replacing the dependent constraints with independent ones, a much simpler constraint structure has been obtained which gives the opportunity of the deeper analysis of the dependencies between events and control actions in the network. The effectiveness of the proposed control technique is shown by extensive simulations: an average of 48.9% reduction can be achieved in the sum of delays. The proposed method is capable to simulate the effect of individual initial delays which enables us to determine the most critical parts of the railway network in terms of delays.

The general approach - namely the formulation of a scheduling problem as a MILP problem in an MPC framework - could be utilized in other control problems, e.g. the optimal control of complex nonlinear systems by using the piecewise affine approximation of the nonlinear system model.

Chapter 5

Conclusions

In this thesis, optimization based methods were used to analyze and control networked systems in large scale, with complex nonlinear dynamics. New methods were presented both in the topic of the structural analysis of the Kinetic Reaction Networks and the solution of the scheduling problems of traffic networks.

The main effort has been focused on the analysis of the structure of the investigated problems. It has been shown that a proper problem representation can be obtained by exploiting the special structure of problem itself, which leads to the simplification of the emerging optimization problems (e.g. see Figs. 3.7-3.8 and Fig. 4.2 and their interpretation).

In some cases LP problems can be formulated to substitute the original MILP problems and in other cases the MILP structure is transformed resulting in computational tasks that can be solved in reasonable time. Also, it has been shown that the methods can be applied on a parallel architecture.

5.1 New scientific contributions of the work (thesis points)

The new scientific results presented in this work are summarized in this Section. They are arranged in three thesis points as follows.

Thesis I. Numerically efficient algorithms to find sparse and dense realiza-tions of kinetic reaction networks.

I have proposed two algorithms both based on linear programming (LP) having polynomial time complexity to compute dynamically equivalent alternative realizations of a kinetic reac-tion network (KRN). I have showed that with the help of the proposed methods alternative realizations of large scale, biologically motivated KRNs can be computed, too. The algorithms are compared with the mixed-integer linear programming (MILP) based algorithm available in the literature and the correctness of the solutions is shown. I have also concluded that the introduced new methods outperform the MILP-based solution in terms of the time consumption of the solution. (Section 3.2)

Corresponding publications: [1, 6]

Thesis I.a

I have proposed an LP-based algorithm to compute dynamically equivalent realizations of a KRN containing minimal number of reactions. The so-called sparse realization of the reaction network is computed via the column-wise L1-norm minimization of the off-diagonal elements of the Kirchhoff-matrix. (Section 3.2.1)

Thesis I.b

I have proposed an LP-based algorithm to compute a dynamically equivalent realization of a KRN containing maximal number of reactions which is proven to contain all possible realizations of the reaction graph as a subgraph. The method to compute the so-called dense realization of the reaction network is based on the relaxation of the MILP-based method known from the literature: the column-wise sum of the introduced real-valued auxiliary variables corresponding to the off-diagonal elements of the Kirchhoff-matrix has been maximized.

(Section 3.2.2)

Thesis II. New methods to compute weakly reversible and mass conserving realizations of kinetic reaction networks.

I have proposed new methods to compute dynamically equivalent and linearly conjugate alternative realizations of a kinetic reaction network while constraints in terms of the structural properties of the reaction graph and/or dynamical properties of the described system are present, too.

Corresponding publications: [2, 4]

Thesis II.a.

A new, linear programming-based method with polynomial time complexity is proposed to compute linearly conjugate, weakly reversible realizations of a kinetic reaction network (KRN). I have compared the method to other linear programming- and mixed-integer linear programming-based algorithms from the literature and it is shown that it outperforms all the others in terms of computational time, hence the algorithm is capable to handle large scale KRNs, too. (Section 3.3)

Thesis II.b.

I have proposed a mixed-integer linear programming-based algorithm to compute dynami-cally equivalent realizations of a kinetic reaction network with mass-conservation property.

The correctness of the results was shown through examples taken from the literature. (Section 3.4)

Thesis III. New solution methods of scheduling problems in traffic networks.

I have proposed a model formulation method for model-predictive controllers which aim to deal with the scheduling problem of railway networks in case of delayed operation. The controller reorders the trains in order to minimize the total delay in the network over the prediction horizon. The model is described with the help of linear constraints. Thus, the controlling problem is formulated as a mixed-integer linear programming (MILP) problem. I have showed the effectiveness of the proposed control technique and a method is proposed for the sensitivity analysis of the model in case of single delays. (Chapter 4)

Corresponding publications: [5, 3]

Thesis III.a.

I have proposed a reordering method of the constraint matrix that can speed up the solution of the MILP problem in the presence of the solver’s preprocessor, too. The method is based on the track-based reordering of the constraint matrix. The track-based reordering means that the constraints corresponding to a given track are collected into one block resulting that the constraint matrix of the emerging MILP problem has a clear block-angular structure. (Section 4.2.1)

Thesis III.b.

I have proposed an algorithm to reformulate the constraints in order to achieve a more simple model formulation. The resulting model shows a clear and simple correspondence between the continuous variables describing the schedule of the events in the network and the binary control variables. Considering these, further analysis of the internal relations of the network model can be done, while problem-specific solution methods can be developed instead of the application of the general-purpose MILP solvers. (Section 4.2.2)