Çekim enerjisi açısından uygun boykesit düzeni ve katar seyir kontrolunun belirlenmesi

Abstract

Konumu belirli iki istasyon arasında bir katarın, önceden be lirli bir seyir süresinde, minimum çekim' enerjisi tüketimi ile ha reketini bir yönde tamamlayabileceği uygun boykesit düzeni ve katar hareket kontrolü incelenmiştir. Biri boykesit düzenini, diğeri katar çekim/fren kuvvetini tem sil eden iki kontrol parametresinden oluşan bu, iki-noktalı-smır değer probleminin çözümünde; Pontryagin'in Maksimum İlkesi ile Ichikawa-Tamura tarafından sınırlı durum uzayında optimizasyon problemleri için geliştirilen koşullardan yararlanılmıştır. Bölüm 2'de hamilton fonksiyonundan elde edilen regüler çözüm lere göre boykesit düzeninin, maksimum eğim değerindeki iniş-çıkış özellikli iki kesimden oluştuğu ve katar hareket evrelerinin de maksimum ivmeli hızlanma, frenlemeye hazırlık ve maksimum ivmeli frenleme biçiminde sıralandığı belirlenmiştir. Kısıtlı durum uza yındaki çözümler sonucunda da boykesit düzeninin iniş-çıkış kesim leri arasında, rejim evresinde geçilen üçüncü bir tek eğimli kesi min yeraldığı kanıtlanmıştır. Bölüm 3'te çekim ve fren kuvveti sabit alınan bir katar modeli için kapalı-çevrim optimal kontrol analitik olarak belirlenmiş ve optimal yörüngeler grafik olarak gösterilmiştir. Bölüm 4'te Bölüm 3' teki çözümlerden ve sonuçlardan yararlana rak minimum seyir süreli optimizasyon problemi çözülmüş ve bir de miryolu güzergahının etüdünde uygun boykesit tasarımına ilişkin diğer optimizasyon sorunları incelenmiştir. Bölüm sonunda, boyke sit düzeni belirli bir güzergah için kapalı-çevrim kontrol denklem leri verilmiştir. vııı

In general, the 15-20 percent of the total energy consumption is consumed in transportation sector. Railroads and waterways are' inherently, more economical than highways and airways in terms of energy consumption. Energy is consumed, partly, during the haulage as tractive energy or motive power, and partly at the stations and other fixed installations in railroads. Among the factors which influence the tractive energy consumption are the characteristics of motive power, vehicles, traffic and the conditions of track (gradients, curves, tunnels, gauges, etc.). It is possible to reduce tractive energy consumption of a train, not only by choosing available vehicles and economical speed, but also by applying more efficient train operation. Furthermore, the geometrical shape of a track profile or the tunnel trajectory of a metro network affects the tractive energy consumption due to gravity traction. This study deals with the problem of determining both the tunnel trajectory, or the shape of the profile of a track, as well as the control of the operational modes of the train in order to minimize the tractive energy consumption between two successive stations. This is, mathematically, a two-point-boundary-value control or optimization problem containing two control parameters. These parameters represent the slope of the track, and the tractive/ braking force. The Pontryagin's maximum principle and some other variational techniques are used to solve this problem, This study is divided into the following chapters Chapter 1 : Introduction Chapter 2 : Mathematical Formulation of the Control Problem and Determining of Optimal Controls by using the Maximum Principle t Chapter 3 : Optimal Trajectories which Minimize the Tractive Energy of a Train Running with Constant Tractive/ Braking Force between Two Successive Stations Chapter 4 : Optimal Controls and Trajectories which Minimize Running Time of the Train Examined in Chapter 3 and a Brief Review of Other Optimization Problems ixConclusion and Appendices In the first chapter, studies concerning the problem are summarized and examined in three parts. In the first part, operational expenses, construction costs and energy consumption are considered in relation to grades and tunnel trajectory (including cyclo train gravity drive). The second part is directly related to the optimal train operation which minimizes the tractive energy. The third part includes the same control problem adopting a heuristic approach for tunnel trajectory and is given an iterative numerical solution by using a direct search method. At the end of the chapter, the problem examined in this study is defined and some notions are given about the solution method which is a variational technique, In the second chapter, the optimal control formulation is estab lished by assuming that the geometrical shape of the profile is a chain-curve consisting consecutive chords. The slope of a chord represents the grade of that track segment. Some notions and remarks on the formulation are as follows : i) The following control variables are defined by choosing the distance travelled as the parameter : u1 (x) : slope of a segment (or grade) u` (x) : tractive/braking force ii) The state variables are as follows : x1 (x) : ordinate of the trajectory x` (x) : travel time of the train x_ (x) : velocity of the train iii) The physical constraints on the control and state variables (or inputs) are as follows : (un). < u. < (u. ) 1 mm 1 1 max (u_). < u0 ^ (u_) 2 mm 2 N 2 max where (u`). and (u2) are the maximum deceleration and acceleration due to braking and tractive forces. The speed of the train is limited as 0 < x_(x) < v ¦i m where v is the maximum velocity, m J iv) The initial and terminal boundary conditions are given as follows :xx (o) «== o x` (o) ¦» O x- (o) «- O xx (f ) - ± h x2 (f ) = tf x3 (f ) - O where ± h is the difference in elevation of the stations, t-, is the specified arrival time, v) The performance measure to be minimized is : x. 2 (U2 ~ K' dx) ' x where x and x£ are the initial and final positions of of the state, vi) It is assumed that the equation of train motion is expressed by a concentrated mass approximation. State equations are established by using the elementary laws of mechanics. The optimal controls which maximize Hamiltonien are determined according to the Pontryagin's maximum principle. The form of the optimal controls are (u0). 2 'mm i max for p` < 0 for 0 < p0 < x0 for x3 < p3 undetermined for Po = 0» P-f x` u` (O. 1 mm for ppx< P3/x3 (u,) 1 max for pp1> p3/x3 undetermined for pp.=p3/x, where p1,p ` are the costate vector components, p is a scalar. Undetermined forms indicate singular solutions or switching points. The regular solutions have the forms of combinations of the controls given above. XIConsequently, the tunnel trajectory or the geometric shape of a track profile is composed of two straight line segments having the steepest possible slopes [(u )., (u1 ) J as regular solutions. For the backward trajectories, tffîPSregulâr optimal controls of train operation which are (U2). » u2=0 (coasting),and (u`) should be applied. 2 max Since the control problem contains the restriction x0<v,. a bounded state variable optimization problem is solved. Jump of the costate vector is presented by Pontryagin et al. as the method to solve the bounded state variable problem. However, the application of this condition is difficult. Hence, the method proposed by Ichikawa and Tamura is used in order to solve the control problem. The method is based on two conditions; the continuity and the branch-off condition. By applying this method, the equation describing the relationship between the admissible controls, by loosing one degree of freedom, is obtained as follows : u2 -w/p - u- «¦= 0 where w is the total train resistance. This optimal control solution on the boundary indicates that when the train speed reaches the maximum velocity j a new operational mode appears and the trajectory has a new admissible segment between the steepest possible slopes which are obtained as regular solution. In the third chapter, optimal trajectories of the train are obtained by solving a two-point-boundary-value optimization problem. The train and the track characteristics are as follows: i) Tractive and braking forces are assumed to be constant. ii) Maximum and minimum slopes of the trajectory are determined by technical considerations such as wheel slip, water drainage, break wear, etc, iii) Train resistance is also considered to be constant. Since the terminal station where the train stops is chosen as the origin, the state equations are restated according to the coordinate system by substituting X = x.-x. Then, the state and the costate equations are solved by applying the optimal control values obtained in the second chapter. For the regular solutions the sequence of the optimal controls are in one of the following forms: U={ (u.), f(u`).,0, (u`) ]}, { (u..)., [(u0).,0, 1 max' LV 2 mm' ' 2 maxJ ' 1 mm' L 2'nun ' (uj ]}, {[(u,), (un). ], (u0). }, {[(u.) 2'maxJ ' L 1 max 1 mmJ ' 2'min ' L 1 max* (un). ],(0)}, {[(u.), 00. ], (u0) } 1 minJ'x ¦ ' L 1 max 1 mmJ 2 max XllThe optimal controls are in the closed -loop form since ithey are obtained as scalars, and no singular solution is occurred. The parametric equations of the optimal backward trajectories and the costate vector components which lie on the boundary are obtained analytically by using the Ichikawa-Tamura conditions. For the graphical representation of the optimal trajectories it is required to construct and examine the (y..,y`,c) three- dimensional space, where y,y,c are the lagrange s undetermined multipliers, since the parametric equations of the optimal trajectories are the functions of y..,y_,c. On the other hand, since the optimal trajectories are defined in the (x.,x2,x3,X) four-dimensional state space, the graphical representation can be shown in cross sections. The equations of the curves in two- dimensional (x1,X) planes are given. The open-loop optimal control conditions are given at the end of this chapter as a special case of the control problem. In the fourth chapter, the optimal controls and the trajectories are obtained so as to minimize the running time of the train. The performance measure is given as follows : dx/x3 The optimal controls that maximize the Hamiltonian Function, in the form of the bang-bang concept, are [(u~)., (u_) J and [(u )m.n, (Ul) J controls related! -i. ~.-- ` i' ».. j * mm ' 2 maxJ, ^. J. Any singular solution does not arise and the controls relateâto bounded state space are the same as those obtained in the minimum-energy problem. By integrating the differential equations of the necessary conditions for optimality, the parametric statements of the optimal trajectories and the costate vector components are found. The graphical representation of the trajectories is easily obtained, because the minimum-time problem is a special case of the minimum-energy problem with no coasting mode (u` =» 0). The design procedure of a track profile (or a tunnel trajectory) is illustrated for a new construction according to the minimum-energy and the minimum-time criteria by using the results obtained formerly. The problems which can be arise in the location study of a new line are examined in the preceeding section s of this chapter. Another control problem to be dealt with at the end of this chapter is the determination of the control conditions of a train traveling on a existing line, This solution gives the parametric equations of the optimal trajectories of the train modes of operation. XlllTheoritical and technical results of this study are given in the conclusion section, Pontryagin's maximum principle and the conditions for the bounded state variable problems are presented in the Appendices A and B, The state equations of the mathematical formulation of the control problem which is given in the second chapter is examined in the Appendix C, XIV