Models for Conductor Size Selection in Single Wire Earth Return Distribution Networks Geofrey Bakkabulindi Mohammad R. Hesamzadeh Izael P. Da Silva Department of Electrical Engineering and Mikael Amelin Strathmore University Makerere University Department of Electric Power Systems Nairobi, Kenya Kampala, Uganda Royal Institute of Technology (KTH) Email: [email protected] Email:[email protected] Stockholm, Sweden Email: [email protected] [email protected] Abstract—The use of the ground as the current return path of the full Carson model, are presented in [7] and [8]. often presents planning and operational challenges in power This study considers the aspect of conductor selection in the distribution networks. This study presents optimization-based planning of SWER systems. The SWER line model, based on models for the optimal selection of conductor sizes in Single Carson’s line, is used to develop optimization-based conductor Wire Earth Return (SWER) power distribution networks. By size selection algorithms for SWER overhead lines. Two using mixed integer non-linear programming (MINLP), models models are proposed: the first determines the optimum SWER are developed for both branch-wise and primary-lateral feeder selections from a discrete set of overhead conductor sizes. The line conductor size for each network branch, and the second models are based on a mathematical formulation of the SWER selects the optimum overhead conductor sizes for the primary line, where the objective function is to minimize fixed and variable and lateral feeders respectively. Both models are formulated costs subject to constraints specific to SWER power flow. Load subject to load growth over different time periods. They are growth over different time periods is considered. The practical tested using a case study extracted from an existing SWER application is tested using a case study extracted from an existing distribution line in Namibia, and subjected to a sensitivity SWER distribution line in Namibia. The results were consistent analysis that considers different network operating scenarios. for different network operating scenarios. The rest of the paper is organized as follows: Section II I. INTRODUCTION presents the mathematical model of the SWER line. Section III presents the summarized formulation of the SWER power flow THE importance of electricity in enhancing economic model. Section IV introduces the two models formulated for development and social welfare is well established. Whereas selecting the optimal overhead conductor for a) each branch, the electrification of urban areas is generally widespread, and b) the primary and lateral feeders of a SWER network. electrifying rural areas has always presented technical and Section V gives the numerical analysis results of the proposed economic challenges. The major setback is the high cost models and the conclusion is given in Section VI. of connecting relatively small and sparse rural loads to the medium voltage (MV) network [1]. II. THE SWER LINE MODEL Single Wire Earth Return (SWER) distribution systems supply single phase power to rural areas economically from The model of the SWER distribution line is based on the main MV grid network, whereby the earth forms the Carson’s model for overhead transmission and distribution current return path [2]. The technology, initially proposed by lines that include the effects of earth return [5]. Carson’s line is Mandeno in [2], has proven to be cost-effective in electrifying used to determine the self and mutual impedances of overhead rural areas with small and scattered loads in countries such conductors with earth return. In the model, Carson considers as Australia, Namibia, etc. [3]. The lines are typically spur a single overhead conductor of unit length parallel to the earth extensions from radial three phase feeders of 11, 22 or 33 kV and carrying a current with the return path underneath the supplying single phase power at 12.7 or 19.1 kV, often via an earth’s surface. isolating transformer [4]. The earth itself is modeled as a single return conductor of The planning of SWER distribution systems faces different infinite length and uniform resistivity with a geometric mean challenges compared to conventional systems due to the earth radius (GMR) of 1 m [5], [6]. The distance, Dag, between the return circuit. Solutions to the problem of earth return power overhead conductor and earth return path is a function of soil flow modeling were first proposed by Carson in [5]. In resistivity: higher soil resistivity causes return current to flow Carson’s model, the earth is replaced by a plane homogeneous deeper from the earth surface increasing Dag, and vice versa semi-infinite solid conductor and modeled accordingly [6]. [6]. Details of the full development of the Carson line model Other approaches to ground modeling, mostly simplifications are given in [5]–[7]. For brevity, only the relevant impedance 978-1-4673-5943-6/13/$31.00 c 2013 IEEE V V Z Z J equations for single wire earth return networks are given here. ja;t = ia;t − aa ag la;t 8l 2 M; t 2 T The SWER overhead line impedance, Zaa, is given by (1). Vjg;t Vig;t Zag Zgg Jlg;t (8) Zaa =z ¯aa +z ¯gg − 2¯zag (1) where Iia;t and Iig;t are the current injections at node i for the t S where z¯ is the line self-impedance, z¯ the ground self- overhead line and earth return in period respectively, ia;t is aa gg i t V impedance, and z¯ the mutual impedance between the line and the specified complex power load at node in period , ia;t ag V i earth, all of which are defined by (2) through (4) respectively. and ig;t are the complex voltages at node for the overhead t Y The factor (z¯ − 2¯z ) in (1) represents the impedance conductor and earth return in period respectively, ia is the gg ag i N T correction due to the earth presence [6]. shunt admittance at node , is the set of all network nodes, is the set of time steps in the full planning period, and i and j −4 2ha are the incoming and outgoing nodes of branch l respectively. z¯aa = ra + j4π · 10 f ln (2) GMRa The branch impedances are as calculated in (1) to (4). All parameters and variables in (5) to (8) are complex. 2 −4 −4 z¯gg = π · 10 f − j0:0386 · 8π · 10 f + j4π IV. CONDUCTOR SELECTION PROBLEM FORMULATION 2 (3) × 10−4 · f ln The objective of conductor selection is to choose the 5:6198 · 10−3 conductor size with minimum investment costs and power losses subject to voltage regulation, load flow, and current −4 ha z¯ag = j2π · 10 ln (4) carrying constraints [9]. The study considers a single-phase pρ/f SWER isolating transformer supplying power from a MV three-phase network to rural loads at minimal cost with load where r is the resistance of the phase conductor a (Ω/km), a growth in different time periods. The output terminals of f the frequency (Hz), h the height of the conductor a above a the isolating transformer were considered to form the infinite the earth (m), GMR the geometric mean radius of a (m), and a bus [10]. Therefore, the transformer’s installation costs and ρ the ground resistivity (Ω·m). internal losses were excluded from the analysis. In addition, When load current is conducted into the earth, dangerous the following assumptions were made. touch and step potential gradients can result for both man and beast [4]. However, careful design of the earthing system 1) The network is supplied by one SWER isolating ensures that these voltages are kept within safe levels. For transformer located at a known point of grid extension. safe SWER system operation, the earth current should be 2) Load data including size, location, and estimated load limited to 25 A at 19.1 kV under typical conditions and to growth rate are known beforehand. 8 A where the SWER lines are likely to interfere with open 3) Only the peak load is considered for each year of the wire communications [3]. Further details on SWER earthing planning period and there is no unserved energy. requirements can be found in [3]. 4) Equipment data including unit costs, capacities, and electrical characteristics are readily available. III. SWER POWER FLOW MODEL A. Constraints for Branch-wise Conductor Selection The SWER load flow formulation is based on the for- ward/backward sweep method presented in [6] for earth return It is required to choose an optimum conductor size for networks. All nodal current injections due to loads and shunt each branch from a finite set of conductor options whose elements are first computed based on initial voltage values parameters of resistance, reactance, thermal limit, and unit for both the overhead conductor and ground return path using fixed cost are known. Therefore, optimization techniques were (5). The branch currents are then calculated using Kirchoffs formulated in (9) through (14) to select the parameters from Current Law (KCL) in (6) and (7) for the line and ground each corresponding finite set that give the optimum conductor. respectively, where the loads are represented by their equiva- The chosen parameters were then incorporated into the rest of lent current injections. According to the KCL, the sum of all the constraints (15) through (19) and objective function (31). branch currents entering and leaving a node is equivalent to X r0 = y · r 8l 2 M; t 2 T (9) the load current at that node. Finally, nodal voltages for both laa lct laac overhead line and earth return are updated using (8). This leads c2CT X to an iterative procedure that ends when the difference between x0 = y · x 8l 2 M; t 2 T (10) the specified and calculated current injections at each bus is laa lct laac c2C minimum.