The main purpose of this paper is to elaborate a calculation program in Pascal language, using Delphi environment. This calculation program is designed to solve the power engineering optimization problems using the method of optimal Hamiltonian path and circuit, solved with the so called "Latin multiplication". For illustrating the use of the algorithm and the calculation program we present a problem of optimal equipment programming for realizing an ensemble of electrical overhead lines.This paper is structured in 4 parts. In the first part of the paper we present the application as a problem of optimal Hamiltonian path and circuit. In the second part of the paper, we determine the optimal Hamiltonian path (circuit) in a graph using the so called Latin multiplication method. In the third part of the paper we present a representative numerical application. In the fourth part of the paper it is described the calculation program

Constantin Bărbulescu

Gabriel Limbean

Marius Groza

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Constantin Bărbulescu, Marius Groza, Gabriel LimbeanOptimal Equipment Programming for Realizing an Ensemble of Electrical Overhead LinesThe main purpose of this paper is to elaborate a calculation program in Pascal language, using Delphi environment. This calculation program is designed to solve the power engineering optimization problems using the method of optimal Hamiltonian path and circuit, solved with the so called "Latin multiplication". For illustrating the use of the algorithm and the calculation   program  we   present   a   problem   of   optimal  equipment programming for realizing an ensemble of electrical overhead lines.This paper is structured in 4 parts. In the first part of the paper we present the application as a problem of optimal Hamiltonian path and circuit. In the second part of the paper, we determine the optimal Hamiltonian path (circuit) in a graph using the so called Latin multiplication method. In the third part of the paper we present a representative numerical application. In the fourth part of the paper it is described the calculation program.1. Presenting the problemLet us consider an enterprise specialized in constructing high voltage overhead power lines, using for this purpose a set of utensils. The transport of those utensils from one building site to another generally implies high costs. The transport expenses consist of a fix part and a variable one. The fix part is linked to removing of the utensils from the device and bringing them to the embarking station, as well as entering the new device in the arrival station. The variable part consists of the cost for the transport on the railroad.We are required to establish the order in which we will execute the power lines in order to minimize the cost of transport of the utensils.Additionally, we establish the following:a) the cost of transport of the utensils on the railroad is proportional to the distance  of  transport, the minimization of the cost being equivalent to the minimization of the distance of transport;b) the utensils are received on the initial point of the power lines and are shipped to the final point of the lines, which leads to the conclusion that generally 39ANALELE UNIVERSITĂŢII “EFTIMIE MURGU” REŞIŢAANUL XII,  NR. 1,  2005,  ISSN 1453-7394the distance from the i  building site to the j  building site is not the same as the one from j  to i.c) to begin the approximation, we consider that in the initial moment the utensils may be found on any building site, and in the end we state the initial building site; in both alternatives the final building site may be any of the building sites, instead of the initial one.Let us consider that the building sites of the power lines are the nodes of a graph, and that the railroads between them are the arcs of the graph (the transport lengths are the values of the arcs), then the solving of the problem is reduced to finding a  Hamiltonian path (HP) of minimum value, which have the initial node corresponding to the initial building site.The problem may be regarded from the point of view of returning with the utensils in the initial point of departure, in this case being necessary to find a Hamiltonian circuit (HC), of minimum value.Analogous mathematical models may be obtained in other situations, leading to the necessity of knowing the Hamiltonian paths or the minimal value HP:a) a certain system must go through a certain number of states; knowing the costs of going through states, we are required to determine the optimal succession of the states; b) setting the order of launchings in production or repartitions of certain marks in the workshops of the mechanical and power services, knowing the costs of going from one mark to another .2. Determining the optimal Hamiltonian path and circuit via the Latin multiplication Via a special kind of multiplication, the Latin multiplication, we may successively find all the elementary paths of  1 n , 1 -  length, those of n-1 length (if they exist) being the Hamiltonian paths.To the G   graph we assign the Latin matrices  [ ]) 1 ( M  and  [ ]) 1 (M defined as follows:[ ]ïîïíì=¹ Ï¹ Î=j i if 0j i , A ) x , x ( if 0j i , A ) x , x ( if x xM j ij i j i) 1 (;[ ]ïîïíì=¹ Ï¹ Î=j i if 0j i , A ) x , x ( if 0j i , A ) x , x ( if xM j ij i j) 1 (.(1)40We define the Latin multiplication of the matrices  [ ]) 1 ( M  and  [ ]) 1 (M , noted as follows: [ ] [ ] [ ]) 1 ( ) 1 ( ) 2 ( M M M L = (2){ } j l il j l illlj il) 2 (ij s s 1 1 m m ,......, m m m m m = =  (3)where s < n, and the reunion does not contain any zero terms nor any terms for which the sum  å=n1 llj il m m contains repeated letters. Then j l i x x x1 ,  j l i x x x2 , .......,  j l i x x xs  are all path of elementary length 2 which unite the xi node with the xj node.Analogously we define the Latin matrices ) k ( ) 3 ( ) 2 ( ] M [ , , ] M [ , ] M [ [ ] [ ] [ ]) 1 ( ) 1 k ( ) k ( M M M L- = (4)( ) ( ) ( ) { } j l1 kil j l1 killlj1 kil) k (ij s s 1 1 m m ,......, m m m m m- - - = =  (5)where s < n, and the reunion does not contain any zero terms, nor repeated nodes.The elements of the  [ ]) k ( M  matrix will give all the elementary paths of k length of the G graph. Finally, the elements of the  [ ]) 1 n ( M-  matrix will give all the Hamiltonian paths of the graph (if they exist).If we are only interested in the HP, then the volume of calculation may be considerably  reduced by introducing the   [ ]) k (M matrices, which may be obtained from  [ ]) k ( M  by cutting off the first peak of each element (elementary path of k length). The following relation may then be written, which, applied accordingly, to n, may significantly reduce the calculations: [ ] [ ] [ ]) 1 ( ) 1 k ( ) k ( M M M =- L (6)In order to apply the abovementioned method, the following steps must be observed:a) writing the Latin matrices  [ ]) 1 ( M  şi  [ ]) 1 (M  of the G  graph;b) by corresponding successive Latin multiplications we will obtain the Latin matrix [ ]) 1 n ( M- , and its elements are all HP (if there are such paths in the G graph);c) calculating the HP value, in the end selecting the optimal HP (of minimal value).The Hamiltonian circuits are determined calculating the Latin matrix   [ ]* ) n ( M , defined by the following relation[ ] [ ] [ ]) 1 ( ) 1 n ( * ) n ( M M M L- = (7)41In order to obtain the HC (if it exists) for every element of the  [ ]) 1 n ( M-matrix we assign as initial element the peak corresponding to the line index of the element.3. Numerical applicationLet us consider an enterprise specialized in constructing EOL, which, during a year, has to produce 6 high voltage EOL, using a set of utensils. The distances of transport between the 6 building sites are known and presented in the  [ ] C  matrix of values. The elements  ij c  of the [ ] C  matrix represent the distances between the building sites, in km. We are required to set the order in which the EOL are to be executed in order to minimize the cost of transport for the utensils. 1 2 3 4 5 61 0 625 604 467 389 2972 486 0 876 541 888 3443 480 653 0 569 330 3384 257 672 710 0 196 5805 345 734 424 282 0 6856 220 211 542 537 416 0We will now present two alternatives of applying the Latin multiplication method, which  differ through their way of defining the partial graph to which the Latin multiplication method is applied.First, let us consider an asymmetrical partial graph (from the two “symmetrical” arcs,  we will store the one with the lowest value as being part of the partial graph). For this graph we will write the Latin matrices  [ ]) 1 ( M  and  [ ]) 1 (M , and we will do the following operations:42251 34 6Figure 1.[ ] [ ] [ ]) 1 ( ) 1 ( ) 2 ( M M M L =[ ] [ ] [ ]) 1 ( ) 2 ( ) 4 ( M M M L =[ ] [ ] [ ]) 1 ( ) 4 ( ) 5 ( M M M L =The elements of the [ ]) 5 ( M  matrix will give all the HP.12345612341616[ ] =) 1 (M456361626465534131322422115161654[ ] =) 5 ( M25436245365243452116235224455224645245245545664[ ] =) 1 ( M3512121 41621 64145154451 61651121 4151411521321 34151351  3611621465241512624516452133 51453164245441543[ ] =) 2 (M234566526246455244523453243525612456213122124[ ] =) 1 ( M33132343536441455515266162646536541352413624132451364516564623653643624515 1 164532 162453 153624    165324 145362444232145125452545343536326123456336452111234566[ ] =) 4 ( M264253[ ] =) 2 ( M224111234562412[ ] =) 1 (M4561126The second alternative requires the segmentation of the partial graph from the first altern145362 15326445ative, by introducing also the arcs for which cji has a value equal to   or just a bit higher than cij, but relatively low when compared to other elements of the [ ] C  matrix: 1-4 (467 compared to 257), 1-5 (389 comp46ared to 345), 1-6 (297 compared to 220), 2-6 (344 compared to 211), 5-3 (421 compared to 330) and 5-4 (282 compared to 198). After writing the Latin matrices [ ]) 1 ( M  and [ ]) 1 (M  we do the Latin 47operations. The [ ]) 5 ( M  matrix contains   a number   of 82 HP, being synthesized   in the next table those which value is under 2000, noticing the fact that also exists paths between the peak 1 and the peak 2,   3, 4,   6, the 48peak 2 and 1,   3, 4,   6, the peak 4 and 1,   2, 6, the peak 5 and 1,   2, 4, peak 6 and peaks 1,   2, 3. The HP marked   in the table are the optimum ones for the situations when it   is imposed the initial peak, the solution 49„optimum optimorum” being 3-5-4-1-6-2, having the value 1377.3652413624512264531265341245361216453241653261453264153216534265314261534215364245316241536214536[ ] =) 5 ( M 3364521352641326451345261326541365241362451341652361452364152316452345162354162352164    316524365214    361524352614    362154351624    31526432165432615432651432641532614532164536241531624536214532416535241632451634521634152631452644536214532614153624165324531624532164153265536241532641534162 532416664532162453165324164153261453250Table 1.123456453621165413216453191025181936214517001124153636621453178319372319593261451882926153434624531185118881145362163514362451163327453261183621624531668153541621377284531621607315362419001636145219572941536216164264531198017345162162030536241176261952214536624153241727245316614532221736HPValue Nr. HP ValueThe HC is determined based on the relation: 19393265411237194631624510351943362415Nr.[ ] [ ] [ ]) 1 ( ) 1 n ( * ) n ( M M M L- =(8)For  n =   6 we obtain the Latin matrix[ ]* ) 6 ( M,which is a diagonal matrix. By 518524536117181835162417243153416217556241653193219362154170632536214Nr.analyzing the [ ]* ) 6 ( M  matrix,   we come to the conclusion that there are   7 HC, the values of which are presented in the following table, the optimal cycle being 1-4-5-3-6-2-1   , with the value 2121 (second line in the table). We notice 52the existence of other two HC with values close to the optimal   HC (the third an the fourth line of the table)335316241950826415319482119237261453165020326154HPValue1645321 2416532 3264153 4536214 5326415 64532161453621 2614532 3624153 4532614 5362415 6245316[ ] =* ) 6 ( M 1624531 2164532 3261453 4165324 5326145 65324161536241 2641532 3162453 4153624 5362145 64153261653241 2415362 3621453 4532164 5324165 62145361453261 2145362 3241653 4153264 5321645 624153653Table 2.Nr. Circuit Value Nr crt Circuit Value1 1645321 2592 5 1532641 26012 1453621 2121 6 1553241 25853 1624531 2148 7 1453261 23034 1536241 2157Both ways to apply the Latin multiplication method are approximate methods (due to the reduction of the initial graph), with great chances for the second alternative to find the exact solution of the problem. Also, the second alternative allows determining HC close to the optimal solution or even the optimal solution itself, as in the present case.4. LATIN calculation program4.1. OverviewThe CRITIC calculation program has been elaborated in Turbo Pascal language, using the Delphi environment. It finds the CP via a program graph and it solves a series of related problems.The following aspects related to the calculation program may be noted:a) The program solves all aspects related to the creation and actualization of the data base related to a certain application, as well as the loading and saving of the files that contain the data bases;b) A large number of check-ups for the compatibility and the accurateness of the elements in the data base, with warning or error messages if necessary;c) The partial graph may be set by the program or by the user;d) The problem of the optimal HP is solved via the Latin multiplication method, using  special  storing   techniques   for   the   sparse   matrices,  with   messages according to each case that may present itself during the solving process; e) The problems also having a didactic side, the visualization of the results may be  done according to the user’s wish: from the sole display of the final results to the visualization in a work window of all the intermediate results for each iteration, with the option of browsing them;f) The option of saving or listing the result files is available.4.2. Delphi®  environmentIntroduced as Object Pascal 7.0, the firm that produced it was not able to give up the codename Delphi, due to its market popularization.54Delphi introduces several concepts unknown in the Pascal language, such as: classes, properties, interfaces and objects. The classes derive from the old objects Pascal, the method concept setter-getter being introduces in an original manner.Delphi is part of the RAD (Rapid Application Development) applications package, being a visual environment, via which we can easily build Windows interfaces.The programming style is named OOP (Object Orientated Programming), known in the world of Borland developers as PME (Properties-Methods-Events).There is a great similarity between the Delphi environment and Turbo Vision, both operating new objects and having a new way of treating events and objects. Due to the fact that Windows is a system based on messages, the Pascal should be modified  so as to respond to messages, in order to comply with the present requirements. Delphi introduces the concept of notification procedure, which is actually a method of a class, this method is indirectly called by the system, its filling being actually a perpetual waiting after a user’s action.At this time, worldwide, for the standard of home user, the operating system is minimum Windows 98, and the industrial standard is Windows NT4. A strong tendency of migration to the NT engines is obvious, now existing the possibility for a simple user  and a firm to use the same operating system (Windows XP) for different problems in order to achieve maximum efficiency. This operating system is promoted in all fields where a computer is required.The integrated developed environment (IDE) Delphi has been chosen since Pascal is an easy language, which allows introducing easily and in a graduate manner all  the Windows programming problems, without losing much of the specifics of the applications. 4.3. Using the programLATIN  program, elaborated in  Delphi®  environment, version 5, is a classical application for Windows®, offering a facile interface for the students, during the practical works  for the  Techniques of optimization in the power industry course, as well as for all those interested in such applications.This third version is developed using the Delphi environment, for IBM-PC computers  for the Windows® 98 engine. Delphi represents and integrated environment   of  development,   offering   a   complete   set   of   application   for developing the “standard” Windows® application, based on the Pascal language, with the extensions offered by the Object Pascal.The applications covered by the Delphi environment range from the office applications and / or utilitarian (Office Ready), it offers an intuitive simple base for the standards COM, DCOM, OLE, dB, RDBM, XML, HTML, MIDAS, to the servers of distributed applications. The program package covers the following objectives:· solves optimization problems for the power industry55· offers a didactical base for the works for the practical works for the Techniques of optimization in the power industry course.When it is launched, the program window will have the following aspect:Figure 2.A new file (empty) is loaded, and it contains a matrix which represents a graph with 7 nodes and which is to be filled in by the user with all the necessary data. This is the value matrix. The program also allows in this window the option to see the Boolean matrix corresponding to the truth values regarding the existence (value 1) or not (value 0) of a path between 2 nodes.Before initialising the process of filling-in the elements of this matrix, the user must access the menu Vedere – Optiuni, or o click on the according button on the toolbar. This command will lead to another window, in which the user may specify the number of nodes of the graph, as well as other figures (the number of exact digits, the initial value of the data etc.).Figure 3.  Figure 4. After introducing the information necessary in order to fill in the matrix, the user has the option to save that file under a certain name, for a subsequent use.The window of the application for the matrix that contains the user data and the table with the existing arcs will have the following aspect.56After having gone through the process of filling-in the data matrix, the user accesses the menu Calcul – Calculeaza.Figure 5.Figure 6. After selecting this option, the “Optiuni” window (seen earlier) will appear again and it will mainly allow to configure 3 parameters.· the dimensions of the problem (tackled above);· the   display   options:   whether   to   print   the   initial   data   or   not,   the intermediary results, whether to clear or not the screen before displaying the results;· the calculation options: whether the partial graph is generated by the program, or set by the user. The user is also allowed to access several options to modify the partial graph in case the latter is generated by the program.Figure 7. Figure 8.57By clicking the OK button, the calculation process is initiated. In the window which appears upon completion, the results of the calculation process are presented, according to the user’s wishes. The application allows the export of the initial data, respectively of the results as text or rtf files (fig. 9), files that may be later opened via programs in the Office® package.Figure 9. References[1] Kilyeni   Şt.,  Tehnici   de   optimizare   în   ingineria   energetică.   Editura Orizonturi Universitare, Timişoara, 2004[2] Kilyeni Şt., Metode numerice. Algoritme, programe, aplicaţii în energetică, Ed. a 3-a.  Editura Orizonturi Universitare, Timişoara, 2004[3] Kilyeni Şt., Groza M., Limbean G., Tehnici de optimizare în ingineria energetică. Lucrări practice.  Editura Orizonturi Universitare, Timişoara, 2003[4] Momoh J.A., Electric Power System Applications of Optimization. Marcel Dekker Inc., New York, 2000[5] Borland Delphi 3. Object Pascal Language Guide. Borland International, 1997[6] Borland Delphi 3. Visual Component Library Reference. Borland International, 1997Addresses:· Constantin  Bărbulescu, student IVth  year,  "Politehnica" University Timişoara, Electrical and Power Engineering Faculty, Bd. V. Pârvan Nr. 2, Timişoara, Romania, e-mail costelbarbulescu@k.ro 58· Asist. Drd. Ing. Marius Groza, "Politehnica" University Timişoara, Electrical and Power Engineering Faculty, Bd. V. Pârvan Nr. 2, Timişoara, Romania, e-mail marius.groza@et.upt.ro· Asist. Drd. Ing. Gabriel Limbean, "Politehnica" University Timişoara, Electrical and Power Engineering Faculty, Bd. V. Pârvan Nr. 2, Timişoara, Romania, e-mail gabi.limbean@et.upt.ro59

Optimal equipment programming for realizing an ensemble of electrical overhead lines

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