Optimal Planning of Radial Distribution Networks by ... - IEEE Xplore

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 2, MAY 2008

Optimal Planning of Radial Distribution Networks by Simulated Annealing Technique Jovan M. Nahman and Dragoslav M. Peric´

Abstract—A method for optimal planning of radial distribution networks is presented in detail based upon a combination of the steepest descent and the simulated annealing approaches. The object of investigation is the complete network of available routes and the optimization goal is to find the routes that provide the minimal total annual cost. The minimum capital cost oriented solution created by applying the steepest descent approach is used as the initial solution for the optimization procedure that is further improved by simulated annealing to obtain the minimum total cost solution. The method takes into account the capital recovery, energy loss and undelivered energy costs. Index Terms—Annual costs, optimal routing, radial networks, simulated annealing.

I. INTRODUCTION LANNING of distribution networks is a very complex task as it implies the consideration of various important issues including costs of different origin, technical and ambient imposed constraints and customer needs concerning the quality of energy delivery [1]–[4]. The goal of an optimization is to find the network solution that satisfies all requirements and constraints with minimum annual cost. The possible network configurations depend on the available routes, source substation and load point locations and on the adopted general concept of their structure. Technical constraints that have to be satisfied for each feeder are the maximum allowable voltage drops at load nodes and the maximum load carrying capacity of lines, at peak load. For typical rural networks with relatively small individual consumers sparsely spread around the source substation, radial networks are preferred to reduce the capital cost. Such network structures are primarily considered in this paper. In order to find the optimal feeder routes various discrete search approaches were used in the past. The optimization of radial distribution networks was conducted using ant colony system algorithm providing the minimum investment and loss cost solution [5]. The potential solutions are conceived based on system experts’ suggestions. The optimization of radial distribution networks was conducted [6] by applying the dynamic programming technique and geographical information systems. The network is formed stepwise by covering the loads beginning from those closest to the source substation and then gradu-

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Manuscript received July 6, 2007; revised November 21, 2007. Paper no. TPWRS-00473-2007. J. M. Nahman is with the Faculty of Electrical Engineering, University of Belgrade, Belgrade, Serbia (e-mail: [email protected]). D. M. Peric´ is with College of Applied Business Studies, Valjevo, Serbia (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRS.2008.920047

ally the more distant loads. The simulated annealing algorithm was applied [7] to find the best solution for further expansion of a radial distribution network through a number of alternative triangular substructures by considering the impact of the reliability associated costs. Branch exchange technique was applied [8] using the minimum spanning tree of the graph of available routes as the initial solution. This solution was then step wise reformed to meet the technical constraints concerning branch loading and voltage drops with minimum capital and loss cost. The optimization method presented in this paper is primarily oriented towards newly planned networks. The best solution is searched for within the graph consisting of all line routes that are available considering the local ambient circumstances. Thus, it uses the widest possible basis for the search. The initial feasible minimum cost solution is determined by applying a steepest descent approach. This solution is further modified step by step using simulated annealing technique to search for the minimum total cost solution including the customer cost caused by load supply interruptions. By an appropriate selection of the graph of routes to be considered, the approach suggested can be simply used for network expansion planning as well. II. NETWORK MODELING A. General The annual total network cost of rural networks considerably depends on the capital cost. The load delivery interruptions and power losses may also be cost effective to some extent due to long supply routes and, therefore, must be taken into account in optimization. As the capital cost is prevailing in most cases, the optimal network configuration is usually radial with a minimal total length of branches. Such a solution is well known in the graph theory as the minimal spanning tree for the graph of the available routes and efficient algorithms are provided for the construction of such trees [9]. Unfortunately, in many cases the minimum spanning tree solutions fail to respect the technical constraints imposed and must be modified through a number of steps to reach a feasible potential network configuration. Therefore, a different specific approach was used in this paper to generate the initial solution, based upon the idea of the steepest descent method [10] applied to minimize the network capital cost. The network capital cost is decreased step by step by removing the most capital cost effective branches among the available set of branches if this removal keeps the feasibility of the solution. A solution is feasible if it satisfies the imposed technical constraints and keeps the connectivity of the network. If this is not the case, the selected branch is kept included and the next most capital cost effective branch among the remaining branches is

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´ : OPTIMAL PLANNING OF RADIAL DISTRIBUTION NETWORKS BY SIMULATED ANNEALING TECHNIQUE NAHMAN AND PERIC

removed. This removal procedure is continued until any further removal of a branch produces an unfeasible solution. As the network during this minimization procedure changes from meshed to radial configurations, general methods for load flow and connectivity analyses should be provided applicable to all possible network structures created in various steps of this process. B. Network Connectivity Check A network graph is connected if there is a path between any two of its nodes. For a distribution network it means that all load nodes are connected to the source node and can be supplied from this node. The graph connectivity can be simply checked [9] by defined as the help of the matrix

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The first iteration for node voltage drops is

(4) All quantities in above expressions are represented by complex numbers. The iteration process can be taken as terminated when the difference in absolute value of voltage drops in two succeeding iterations are not greater than 1‰ of the absolute values of these voltage drops in the last iteration. Currents flowing through network branches are determined from the calculated voltage drops

(5) (1) is the by In (1) is the number of network nodes and adjacency matrix of the network graph with elements being equal to 1 if there is a branch between nodes and and equal to zero otherwise. The network is connected if all off diare equal to 1. agonal elements of C. Load Flow Analysis The load flow analysis is conducted based upon assumption that maximum load currents can be reasonably well predicted for all load points, which is usually the case in network planning studies. The network configurations under investigation are weakly meshed and therefore close to ill conditioned networks. It prevents the use of the bus admittance matrix inversion as a straight way approach to determine the network node voltages from load currents. Therefore, a simple iteration procedure [11] was used based upon the network bus admittance matrix. The network source node is taken as slack bus and the calculated voltages at network nodes are voltage drops by reference to this node. The recursive expressions for iteratively calculating the voltage drops, given in [11], are converted to matrix form

(2) with , , , , designating column vector of voltage drops at network nodes in iteration , column vector of load currents, network bus admittance matrix and diagonal matrix built from the main diagonal elements of bus admittance matrix, respectively. From (2), the following relationship is obtained after elementary manipulations:

(3) with being the unit matrix. Load currents known input variables.

are taken as

is diagonal matrix of network branch admittances where denotes the network incidence matrix. and The iteration process described above has been affirmatively checked for convergence and speed in numerous practical cases analyzed by the authors. D. Reformation of Network Matrices During the optimization procedure the network configuration will be changed by inclusion and exclusion of various branches. Each change in network configuration should be accounted for in the network connectivity and load flow analysis. can be simply reformed using the Adjacency matrix . information contained in the 2 by network edge matrix , of gives the numbers of nodes at the Column ends of branch , say and . If branch is removed from the and in matrix network graph then elements should be set to zero. If this branch should be included in the network the corresponding elements of the adjacency matrix are set to 1. In the load flow analysis the exclusion of any branch is simply modeled by setting

(6) can be used as a single means to completely define Matrix the network graph structure. The direct correlation of matrices and is already mentioned above. Matrix can be too if we adopt that netused to define the incidence matrix are oriented from the nodes in the first row work branches in of the matrix toward the nodes in its second row. E. Network Cost The capital recovery cost equals

(7) with designating the capital recovery rate and being the cost of branch . Cost of branches going from the source substation

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include both the line and corresponding substation costs. By the set of branches in the network configuration under consideration is denoted. In radial networks there are no alternative supply routes and the outage of a branch interrupts the delivery to all consumers supplied through this branch. Thus, the cost of supply interruption can be calculated using the expression (8) designate cost per unit of energy where , , , , and not delivered, load factor, repair duration, branch failure rate, branch current and network rated voltage, respectively. The cost of energy losses equals (9) (10) being cost per unit of energy lost, loss factor with , and and branch resistance, respectively. The total annual network cost that should be minimized equals (11) III. OPTIMIZATION PROCEDURE The optimization flow includes the following steps. 1) Store all data for the complete graph of available routes figuring in relationships (2) and (7)–(10) including madefining graph configuration. Maximum tolerable trix limits for voltage drops and branch loading should also be set. 2) Determine matrices figuring in (1) and (3) from input data. 3) Remove from the graph the most capital expensive branch not considered before. 4) Reform network matrices and check the connectivity of the network created. If the network is connected go to step 5. Else, leave intact the selected branch and return to step 3. 5) Check for the violation of voltage drops and loading capacity constraints. If any of these constraints is violated leave intact the selected branch. Go to step 3. 6) After performing steps 3 to 5 the initial network solution is obtained. Calculate total cost for the solution obtained using (11) and store this value for further reference. Define for the simulated the initial temperature . Set annealing procedure index. 7) Include in the network by chance one of the branches removed in previous steps. Remove by chance a branch from the mesh formed by former branch inclusion. Generate new . Set . temperature 8) Check for the connectivity of the created network. If the network is not connected leave the branch intact and return to step 7.

9) Check for the violation of loading capacities and voltage drops. If there is a violation leave the branch intact and return to step 7. 10) Calculate total cost for the generated network configuration. If this cost is lower than the minimum determined so far, store it and the associated network configuration. Return to step 7 with the created network solution. Else continue with step 11. be the difference between the cost of the best solu11) Let tion so far and the cost of the network generated in the previous step and a uniformly distributed random number. If return to step 7 with the created network solution. Otherwise, leave the branch intact and return to step 7. with being 12) The steps 7 to 11 are performed till the maximum number of optimization iterations selected. In the case of network expansion planning, network sub graphs of available routes for extension should be attached to the graph of the existing network as well as new consumer locations including all necessary data. The optimization flow will generally follow the same pattern as for a new network but the exclusion inclusion search process will be applied to the branches of the added extension sub graph only. The conductor sizes do not considerably affect the capital cost of overhead lines that supply rural areas. Therefore the assumption that all lines have the same conductor size is a rational option, at least in the preliminary planning phase. If different conductor sizes have to be considered, the planning could be expanded in more phases. In the first phase the optimization can be conducted with the same conductor size for all network branches. In the optimal network configuration obtained in the first phase, the branch conductor sizes could be reselected based upon the loads they carry and then the optimization procedure repeated with new conductor sizes. This is an interactive approach as sometimes the solution obtained with new conductor sizes may give higher total cost than before due to the increase of losses and undelivered energy costs. In some cases the generated initial solution could be a meshed network to prevent overloads or high voltage drops. This indicates that changes in conductor sizes should be necessary. A rational solution regarding conductor sizing could also be to adopt for branches connected to the source substation in advance a step higher conductor size than for the remaining network branches. It is worth noting that the optimization procedure described above can be applied to networks containing distributed generation as well. In the majority of cases, the distributed generators decrease the currents flowing through the most loaded network branches and improve voltage conditions. This means that the potentially critical network conditions concerning maximum branch loading and voltage drops occur when these generators are not in operation. As the network can not rely upon a permanent operation of distributed generators, it is reasonable to exclude these generators when checking the feasibility of a network solution. Furthermore, the autonomous operation of distributed generators with no connection with the source substation is not approved by regulations in many countries (including authors’ country) for different reasons. Besides,


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TABLE II LENGTHS OF GRAPH BRANCHES

TABLE III COMPLEMENTARY LINE DATA Fig. 1. Graph of available supply routes.

TABLE I CONSUMPTION AT LOAD POINTS TABLE IV COST AND COMPLEMENTARY LOAD DATA

some types of distributed generators can not operate if not supported by the source substation. From the aforesaid it follows that the distributed generators do not decrease the cost due to the undelivered energy to the consumers caused by network failures. Hence, this cost can be determined by omission of the distributed generation. The total cost caused by line failures should be increased by the production lost cost of the distributed generators. A generator is set out of operation at each branch failure disconnecting it from the source node. These failures can be easily identified using the connection matrix of the network solution under consideration. The distributed generation affects the power losses and, therefore, the network branch currents figuring in(9) should be determined by taking into account the distributed generation. In (2)–(5) the distributed generating sources are modeled as loads with negative real power and appropriate reactive power sign and amount. IV. APPLICATION EXAMPLES A. Case 1 The graph of available network routes for a rural 10 kV network that should be planned is displayed in Fig. 1. There are 24 load points (transformers 10 kV/0.4 kV) and 42 available route segments/branches for their supply from the source 35 kV/10.5 kV substation at node 1. Tables I–IV quote the input data for the example under study. The lines are overhead with Aluminum/Steel conductors with sizes given in Table III.

The substation equipment and building capital cost per outgoing line is 75 k$. This amount is added to all branches directly connected to the source substation. Voltage drop limit at maximum load was taken to be 1000 V. The following parameters have been adopted for the opti, and . mization procedure: Fig. 2 shows the initial solution obtained after performing first five steps of the optimization procedure. The final solution found is displayed in Fig. 3. Table V presents the results of the optimization process. As can be seen, the initial solution is characterized by smaller capital cost but higher loss and energy not delivered costs when compared to the optimal solution. The decrease of loss and undelivered energy costs for the optimal solution is achieved by network branching closer to the source substation than in the initial solution, which reduced the load currents flowing through some heavily loaded branches. However, it is important to stress that the initial solution has provided a low cost option very close to the optimal solution. This can be explained by the fact that the steepest descent procedure, as conceived in this paper, generates feasible network solutions with minimum capital cost that are usually the solutions with minimum total network length. Such networks are in the majority of cases also characterized with low loss and undelivered energy costs. B. Case II Let us consider a 34.5-kV distribution system displayed in Fig. 4. The network, supplied by a 10-MVA substation, feeds an oil production area with 21 load nodes spread over the area,

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Fig. 2. Initial network solution.

Fig. 4. Available routes for supplying the oil production area.

TABLE VI COORDINATES OF LOAD POINTS AND LOADS

Fig. 3. Network minimum cost solution.

TABLE V ANNUAL NETWORK COST US$

Source substations are connected at node 1 (active) and node 2 (back up)

which plane coordinates are given in Table VI. The maximum allowable voltage drops are 3%, all aluminum conductors I/0 are used with parameters as given in [12], capital cost for lines is 10 kUS$/km, power factor of loads is 0.9. The cost of energy losses is 0.05 US$/kWh, loss factor equals 0.35, interest rate is 0.1 and the planning period extends to 20 years. This network was optimized in [5] to achieve the minimum present value cost using ant colony system algorithm (ACSA) and a better optimal result is obtained compared with the result found in [13] where the genetic algorithm (GA) was applied. Both these approaches have not covered the undelivered energy costs.

Prior to the optimization application, the ACSA was tuned to find best ranges for several parameters of the applied heuristic function and for the parameters which should control the optimization process. This was done on the basis of 1000 preliminary program runs. The tuned program was then run 1000 times for the case under consideration to evaluate its performance and to find the optimal network solution. The example described above was also processed using the algorithm suggested in this paper. The same parameters controlling the simulated annealing search as in CASE I are used and no specific tuning to the problem was done. This algorithm has found the same optimal solution as in [5] in a single run.


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and the simulated annealing programming. The minimum capital cost oriented solution created by applying the steepest descent approach is used as the initial solution for the optimization procedure continued by applying the method of simulated annealing. The optimization procedure has shown to be efficient with low consumption of computer time. The method takes into account the total annual cost including capital recovery, energy loss and undelivered energy costs. The approach presented has been applied for optimal new network planning. However, as indicated, it can easily be adapted for the optimization of the expansion of existing networks as well as for networks with distributed generation.

Fig. 5. Optimal network configuration for Case II

TABLE VII PRESENT WORTH COSTS FOR CASE II, US$

This solution is displayed in Fig. 5 and the associated costs are given in Table VII. Total program execution time was 4.578 s (the initial solution was found after 0.281 s). The computation was performed using MATLAB 6.5 code and computer: PC Pentium, 1.61 GHz, 1 GB RAM. The efficiency of the algorithm suggested in this paper can be again attributed to the combination of the steepest descent and simulated annealing algorithms as the steepest descent technique efficiently produces a feasible solution that is very close to the optimal solution. It is, then, in a limited number of steps improved by the simulated annealing algorithm. The minor difference in the energy loss costs for the two optimal solutions stems from the differences in the load flow programs used by these two compared approaches.

V. CONCLUSIONS A method for optimal planning of radial distribution networks is presented based on a combination of the steepest descent

REFERENCES [1] L. Willis, Power Distribution Planning Reference Book. New York: Marcel-Decker, 1997. [2] T. Gönen, Electric Power Distribution Systems Engineering. New York: McGraw-Hill, 1986. [3] E. Lakervi and E. J. Holmes, Electricity Distribution Networks. Stevenage, U.K.: Peregrinus, 1995. [4] R. E. Brown, Electric Power Distribution Reliability. New York: Marcel-Dekker, 2002. [5] J. F. Gomez et al., “Ant colony system algorithm for the planning of primary distribution circuits,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 996–1004, May 2004. [6] N. G. Boulaxis and M. P. Papadopoulos, “Optimal feeder routing in distribution system planning using dynamic programming technique and GIS facilities,” IEEE Trans. Power Del., vol. 17, no. 1, pp. 242–247, Jan. 2002. [7] S. Jonnavithula and R. Billinton, “Minimum cost analysis of feeder routing in distribution system planning,” IEEE Trans. Power Del., vol. 11, no. 4, pp. 1935–1940, Oct. 2004. [8] S. Goswami, “Distribution system planning using branch exchange technique,” IEE Trans. Power Syst., vol. 12, no. 2, pp. 718–723, May 1997. [9] N. Christofides, Graph Theory. New York: Academic, 1975. [10] G. R. Walsh, Methods of Optimization. London, U.K.: Wiley, 1975. [11] G. T. Heydt, Computer Analysis Methods For Power Systems. New. York: Macmillan, 1986. [12] “Electric Power Engineering Handbook,” L. L. Grigsby, Ed., CRC/ IEEE Press, 2001. [13] Y. Da Silva, J. Di Girolano, and A. Ferreira, “Optimizacion de circuitos aereos de distribucion et campos petroleros (in Spanish),” in Proc. I IEEE Andean Region Conf., 1999, vol. II, pp. 967–972. Jovan M. Nahman was born in Belgrade, Serbia. He received the Dipl. Eng. degree in electrical power engineering and the Tech.D. degree from the Faculty of Electrical Engineering, University of Belgrade, Belgrade, Serbia, in 1960 and 1969, respectively. From 1960 to 2001, he was with the Faculty of Electrical Engineering, University of Belgrade, as a Professor in the Power System Department. Currently, he is engaged as a freelance consultant for electric power systems.

Dragoslav M. Peric´ was born in Raca, Serbia, in 1958. He received the Dipl. Eng, M.Sc., and Ph.D. degrees in power engineering from the Faculty of Electrical Engineering, University in Belgrade, Belgrade, Serbia, in 1983, 1989 and 1997, respectively. He is employed at the College of Applied Business Studies, Valjevo, Serbia. His main fields of interest are distribution systems operation and planning and computer applications.