This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE SYSTEMS JOURNAL 1 Convex Models for Optimal Utility-Based Distributed Generation Allocation in Radial Distribution Systems Mohammad Amin Akbari, Jamshid Aghaei, Senior Member, IEEE, Mostafa Barani, Student Member, IEEE, Taher Niknam, Sahand Ghavidel, Student Member, IEEE, Hossein Farahmand, Magnus Korpas, and Li Li, Member, IEEE Abstract—This paper introduces various models for optimal and maximal utility-based distributed generation penetration in the radial distribution systems. Several problems with different probabilistic indices as objective functions constrained by power flow equations, distributed generation penetration, voltage, and thermal limits are proposed to obtain the optimal penetration of distributed generations on rural distribution networks. There are tradeoffs between interests and risks that the distribution network operators or distribution companies may be willing to take on. Thus, to have an effective method for maximal allocation of dis- tributed generations, new indices are proposed, and the problems are formulated as a risk-constrained optimization model. The ob- tained problems have mixed-integer nonlinear programming and nonconvex forms because of nonlinearity and nonconvexity of the optimal power flow (OPF) equations and indices, leading to compu- tationally nondeterministic polynomial-time-hard problems. Ac- cordingly, inthis paper, convex relaxations of OPF are introduced instead of the conventional nonlinear equations. Efficient linear equivalents of the objective function and constraints are introduced to reduce the computational burden. Test results of the proposed models on a radial distribution system are presented and discussed. Index Terms—Convex models, distributed generation (DG), energy losses, mixed-integer programming, quadratic program- ming, risk constraints, voltage profile. NOMENCLATURE A. Indices and Sets E Set of all lines. G Set of generator buses. Manuscript received January 23, 2017; revised July 24, 2017 and November 20, 2017; accepted January 12, 2018. This work was supported in part by the European Union under the Horizon2020 Framework Program under Grant 731148 (INVADE H2020 project). (Corresponding author: Jamshid Aghaei.) M. A. Akbari, M. Barani, and T. Niknam are with the Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shi- raz 7155713957, Iran (e-mail: m.akbari@sutech.ac.ir; m.barani@sutech.ac.ir; niknam@sutech.ac.ir). J. Aghaei is with the Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz 7155713957, Iran, and also with the Department of Electric Power Engineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway (e-mail: aghaei@sutech.ac.ir). S. Ghavidel and L. Li are with the Faculty of Engineering and Information Technology, University of Technology, Sydney, NSW 2007, Australia (e-mail: sahand.ghavideljirsaraie@student.uts.edu.au; li.li@uts.edu.au). H. Farahmand and M. Korpas are with the Department of Electric Power En- gineering, Norwegian University of Science and Technology, 7491 Trondheim, Norway (e-mail: hossein.farahmand@ntnu.no; magnus.korpas@ntnu.no). Digital Object Identifier 10.1109/JSYST.2018.2808197 h (H) Index (set) for hours. i, j, k (N ) Index (set) for buses. s (S ) Index (set) for scenarios. t (T ) Index (set) for days. B. Constants C W Capacity factor of the available wind turbine. C S Capacity factor of the available PV module. EL 0 Energy losses in the base case. F Feeder capacity. γ i (t, h) Importance factor of bus i. H Number of hours. l max,ij Square value of the ampacity of line ij . LC max,ij Maximum capacity of line ij . LL 0 ij (t, h) Line loss in the base case. λ Maximum penetration limit in the system. M Big value for the linearization. N Number of buses. N tap Number of tap positions. P bus,i Maximum allowable penetration in bus i. P s D,i (t, h) Active power of the load connected to bus i. P r W Rated capacity of the wind turbine. P r S Rated capacity of the PV module. P r B Rated capacity of the biomass DG unit. P s W (t, h) Output power of the wind turbine. P s S (t, h) Output power of the PV module. Q s D,i (t, h) Reactive power of the load connected to bus i. r ij Resistance of line ij . ρ s Probability of scenario s. T Number of days. T min Minimum voltage magnitude at the lowest tap position. T tap Step ratio. υ min,i Lower limit of voltage in bus i. υ max,i Upper limit of voltage in bus i. υ 0 i (t, h) Square of voltage magnitude in bus i in the base case. x ij Reactance of line ij . z ij Impedance of line ij . C. Variables BIVR s i (t, h) Binary value for indicating interest voltage rise states in bus i. BLFO s ij (t, h) Binary value for indicating line-flow overload- ing states in line ij . 1937-9234 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.