A Survey of Long Term Transmission Expansion Planning Using Cycles

Transcription

1 Contemporary Engineering Sciences, Vol. 11, 218, no. 12, HIKARI Ltd, A Survey of Long Term Transmission Epansion Planning Using Cycles Pedro Pablo Cárdenas A. 1, Laura Mónica Escobar V. 2 and Antonio Escobar Z. 3 1 Department of Mathematics and GEDNOL Universidad Tecnológica de Pereira, Colombia 2 Department of Electrical Engineering Sao Paulo State University UNESP, Ilha Solteira, SP, Brazil 3 School of Electrical Technology Universidad Tecnológica de Pereira, Colombia Copyright c 218 Pedro Pablo Cárdenas A. et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we present a methodology to solve the problem of optimization the epansion of long-term power transmission networks using a formulation that uses mathematical epressions that are alternatives to the second Kirchhoff law and that are applied to the cycles critical of the system graph. Keywords: Optimization, critical cycles, transmission planning, Kirchhoff s second law 1 Introduction The problem of planning the epansion of the electric power transmission network determines the new investments in transmission lines and substations, which are necessary to allow the adequate transfer of power between the different points of a system for future operation [1].

2 548 Pedro Pablo Cárdenas A. et al. The options associated with this problem are characterized by their high investment costs, as well as the large construction periods and their long recovery times of the investment made. For this type of problems, the planning studies take as a reference the current network and consider the increase in demand in the system nodes, the new generation alternatives and the improvement in power eisting generation, in a time horizon which is usually greater than 1 years. Static planning determines the minimum cost solution from the approach of an optimization problem which considers that the eisting network is part of the future solution, which means that it is not considered the possibility of withdrawing, transferring or leaving disconnected fi elements that are operating in the current network. In this paper a new formulation of the original problem is proposed, replacing the second Kirchhoff law by restrictions associated with the critical cycles of the system, which produce the same optimal solution with less computational effort. The problem is solved initially using the transport model. Then, the eisting corridors who are at their upper limit and the new corridor with additions are identified in the relaed solution. 2 Mathematical model Here, when DC load flow model is used for representing the transmission network and v is the objective function to be minimized, the mathematical formulation is the following [3]: min v = s.t. p,i Ω 1 i,j Ω 1 C ij y ij,k 1 k Ω 2 pi + k Ω2 f pi,k ij + k Ω2 f ij,k + g i = d i 2 i,j Ω 1 f ij n ij fij, i, j Ω 1 3 f ij,k y ij,k fij, i, j Ω 1, k Ω 2 4 g i ḡ i 5 k Ω 2 y ij,k n ij, i, j Ω 1 6 y ij,k 1 y ij,k, i, j Ω 1, k Ω 2, k > 1 7 f ij,k f ij,k 1 M 1 y ij,k, i, j Ω 1, k Ω 2, k > 1 8 ij/n ij f ij,1 M1 yij,1, i, j Ω 1, k Ω 2, n ij > 9 In the previous model, C ij is the cost of adding a circuit in the branch

3 A special case of long term i j between buses i and j. Ω 1 is the set of transmission corridors eisting and Ω 2 the set of new corridors. y ij,k is the binary variable associated with the investment option k of the ij corridor. ij is the base network; fij is the maimum flow allowed for a circuit on the ij path in an eisting corridor. ḡ is the maimum nodal generation vector. n ij is the number of reinforcements added in the ij corridor of the base network, while n ij is the maimum number of circuits that can be added in the ij corridor. M is a parameter defined a priori of large size, which makes the restrictions irrelevant when the variable y ij,k = 1. [3,4] In the previous model 1 it represents the objective function and characterizes traditional planning as a minimum cost problem. The restriction 2 represents the first law of Kirchhoff. The restriction 3 allows to establish the limits of capacity in the eisting circuits and the 4 in the non eisting circuits. Through 5 the limits of generation are established and 6 it establishes the limit of investment. 7 it establishes an order of priority among the investment options and eliminates the eisting symmetry between these options in the traditional disjunctive transport model. 8 and 9 ensure that active power flows are equal in circuits connected in parallel in the same corridor [5,6]. 3 Formulation of the cycles We use below the basic terminology of graph theory, which will guide us to the definition of the cycle. An unguided graph G is a pair V, E, where V is a finite set and E is a family of unordered pairs of elements of V. The elements of V are called nodes or vertices and the elements of E are called paths or corridors of G. Given a corridor between two vertices i, j V, with i = j, we denote this corridor by i, j. Therefore, for a corridor e = i, j E, i and j are called their final points or vertices. In the same way we say that the corridor e is incident to the vertices i and j. Similarly, we say that the verte i is adjacent to the verte j. It is important to note that as we assume an untargeted graph, the adjacency relation is symmetric. The degree of a verte in a non-directed graph is the number of sides incident to it, which we will denote as degi q. A path p of length k, which joins a verte i to a verte j, in a graph GV, E, is a sequence r, r 1,..., r k of vertices such that: i = r, j = r k, with r m 1, r m E for m = 1, 2,..., k. A path is simple if all its vertices are different. In a non-directed graph, a path r, r 1,..., r k forms a cycle if

4 55 Pedro Pablo Cárdenas A. et al. r = r k and r 1, r 2,..., r k are different [7]. A graph G = V, E is a subgraph of G = V, E if V V and E E. Given a set V V, the subgraph of G induced by V is the graph G = V, E where E = {i 1, i 2 E : i 1, i 2 V }. 3.1 Cycle basis Let G = V, E be an unmanaged graph with m sides and n vertices. A cycle of G is a subgraph of G where each verte has an even side. Now, associated with each cycle C there is a value w ij on E, where for any e E, 1, if the link ij C and has the orientation of C w ij = 1, if the link ij C and has opposite orientation to that of C, another case The vector space generated by the incident vectors of cycles is called the space of cycles of G, which has the dimension m n + αg, where m is the number of sides of G, n is the number of nodes or vertices and αg is the number of connected components of G. A maimal set of linearly independent cycles is called a cycle basis. 4 Proposed methodology The idea is to replace the second Kirchhoff law by cycles, that is, θ i θ j = w ij f ij ij =, 1 i,j C i,j C where C is any directed cycle and ij is the reactance in the path i, j. Therefore, the mathematical model for the planning problem of the long-term epansion of transmission systems involving Kirchhoff s second law can be modified using the model given by Eqs.1-9 and enclosing the cycles minimums given by General form of the minimum cycles In an epansion planning problem, given the initial network of the system and future generation data, future demand and characteristics of the options for

5 A special case of long term investment in lines and transformers, the problem is solved using the model of disjunctive transport which does not include Kirchhoff s second law, and is verified in the response if circuits appear in their maimum capacity or circuits added in new corridors. The cycle containing these circuits is determined continously and is a cycle with minimum sum of weights. These cycles, called critical cycles, are added to the transport model and the process is repeated until new circuits do not appear at their upper limit or new circuits are not added in new corridors. When the critical cycles are formed eclusively by eisting corridors, their general form is: i,j C f w ij ij n ij =. 11 ij Now, if one of the cycles involves new proposals, then we must add to Eq. 1 the part corresponding to the disjunctive condition, that is, f w ij ij n ij + w ij f ij,1 ij M z y ij,1, i, j C, ij i,j C 1 i,j C 2 i,j C 2 12 where C 1 represents the subset of links of the critical cycle C, associated to circuits in eisting corridors, C 2 represents the subset of links of the critical cycle C, associated with circuits in new corridors, z is the number of links contained in C 2 to which are associated variables decision binaries y ij,1 [7]. 5 Case study and numerical results To test the proposed methodology, the IEEE 24-Bus reliability test system was employed. A single line diagram is depicted in Figure 1, which consists 24 buses, 41 circuits, a total demand of 8,55 MW, and they can add 5 lines per runner. The optimal solution of the problem of epansion planning presents a cost of investment v = 152,, US.

6 A. MAHMOUDABADI 552 ET AL. Pedro Pablo Cárdenas 123 A. et al. 8. MW 16 MVAR MW 32. MVAR gram of the garver system. ed model for the garver system. ces Power Generation MW Power Loss MW Figure 3. IEEE 24-bus system. Figure 1: IEEE 24-Bus System. Table 4. Quadratic generation cost function. 9 G 1 = G 3 = G 6 = G 1 = G 3 = G 6 = Generator Cost Function $ Capacity MW When applying the proposed methodology to the IEEE system, the following critical cycles are obtained: G 1.14 P P G 2.14 P P When applying the methodology to the IEEE 24-Bus test system, using the disjunctive transport G 7 model,.52 P9 2 + critical 43 P + cycles 781 were 9 obtained, which allow to find the best known solution for this system of 152,, US. Eecution G 13.7 P P time was.2 sec with ticks. G P P G 16.8 P P r the garver system million $. Production Cost Power Losses Cost Total Cost sources is about $ milsults with those reported from in Table 6, the average in- G 18.2 P P G 21.2 P P G 22 P P G 23.8 P P change the trajectory of power flow that may have a severe impact on real power losses. Despite the fact that power losses is increased but total costs has been decreased significantly.

7 A special case of long term Cycle 1 Cycle 2 Cycle 3 15 n n n n n n n n n n n n n 12 = n 12 = n n n = Cycle 4 Cycle 5 Cycle 6 Cycle 7 Cycle 8 f89 n n n n n n n n n n n n = n n = n n n 9 11 = n 1 12 = n n n n = Cycle n n n n n = Table 1: Critical cycles for IEEE 24-Bus System.

8 554 Pedro Pablo Cárdenas A. et al. 6 Conclusion The DC model considered the ideal model to solve the planning problem of transmission epansion can be replaced by the disjunctive transport model plus the set of critical system cycles. In systems of medium and great compleity this shows to be a promising technique to reduce the computation times. Acknowledgements. The authors epress their thanks to the Planning Group in Power Electric Systems and GEDNOL group research of the Technological University of Pereira - Colombia. References [1] A. Escobar et al., Modelos Usados en el Planeamiento de la Epansión a Largo Plazo de Sistemas de Transmisión de Energía Eléctrica, Ed. 1, Universidad Tecnológica de Pereira, 21. [2] V. Laura Monica Escobar, L. Ruben A. Romero, Z. Antonio H. Escobar, R. Ramon A. Gallego, Long term transmission epansion planning considering generation-demand scenarios and HVDC lines, IEEE PES Transmission & Distribution Conference and Eposition-Latin America PES T&D-LA, [3] A. Domínguez, Planeamiento Multietapa a Largo Plazo de Redes de Transmisión Considerando Alternativas HVDC, Pérdidas y Contingencias, 217. [4] L. Garver, Transmission Network Estimation Using Linear Programming, IEEE Transactions Power Apparatus and Systems, , no. 7, [5] A. Escobar, Planeamiento Dinámico de la Epansión de Sistemas de Transmisión Usando Algoritmos Combinatoriales, Diss., Universidad Tecnológica de Pereira, 22. [6] B. Kocuk, H. Jeon, S. Dey, J. Linderoth, J. Luedtke, Xu Andy Sun, A Cycle-Based Formulation and Valid Inequalities for DC Power Transmission Problems with Switching, Operations Research, , no. 4, [7] R. Romero, A. Monticelli, A. Garcia, S. Haffner, Test systems and mathematical models for transmission network epansion planning, IEE Pro-

9 A special case of long term ceedings Generation, Transmission and Distribution, , no. 1, Received: November 1, 218; Published: November 3, 218