Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> This paper gives an approach to implementing a second-order primal-dual interior point method. It uses a Taylor polynomial of second order to approximate a primal-dual trajectory. The computations for the second derivative are combined with the computations for the centering direction. Computations in this approach do not require that primal and dual solutions be feasible. Expressions are given to compute all the higher-order derivatives of the trajectory of interest. The implementation ensures that a suitable potential function is reduced by a constant amount at each iteration.There are several salient features of this approach. An adaptive heuristic for estimating the centering parameter is given. The approach used to compute the step length is also adaptive. A new practical approach to compute the starting point is given. This approach treats primal and dual problems symmetrically.Computational results on a subset of problems available from netlib are given. On mutually tested problems the results show... <s> BIB001 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> Future development of electric power systems must pursue anumber of different goals. The power system should beeconomically efficient, it should provide reliable energysupply and should not damage the environment. At the same time,operation and development of the system is influenced by avariety of uncertain and random factors. The planner attemptsto find the best strategy from a large number of possiblealternatives. Thus, the complexity of the problems related topower systems planning is mainly caused by presence of multipleobjectives, uncertain information and large number ofvariables. This dissertation is devoted to consideration of themethods for development planning of a certain subsystem, i.e.the distribution network.The dissertation first tries to formulate the networkplanning problem in general form in terms of Bayesian DecisionTheory. However, the difficulties associated with formulationof the utility functions make it almost impossible to apply theBayesian approach directly. Moreover, when approaching theproblem applying different methods it is important to considerthe concave character of the utility function. Thisconsideration directly leads to the multi-criteria formulationof the problem, since the decision is motivated not only by theexpected value of revenues (or losses), but also by theassociated risks. The conclusion is made that the difficultiescaused by the tremendous complexity of the problem can beovercome either by introducing a number of simplifications,leading to the considerable loss in precision or applyingmethods based on modifications of Monte-Carlo or fuzzyarithmetic and Genetic Algorithms (GA), or Dynamic Programming(DP).In presence of uncertainty the planner aims at findingrobust and flexible plans to reducethe risk of considerablelosses. Several measures of risk are discussed. It is shownthat measuring risk by regret may lead to risky solutions,therefore an alternative measure - Expected Maximum Value - issuggested. The general future model, called fuzzy-probabilistictree of futures, integrates all classes of uncertain parameters(probabilistic, fuzzy and truly uncertain).The suggested network planning software incorporates threeefficient applications of GA. The first algorithm searchessimultaneously for the whole set of Pareto optimal solutions.The hybrid GA/DP approach benefits from the global optimizationproperties of GA and local search by DP resulting in originalalgorithm with improved convergence properties. Finally, theStochastic GA can cope with noisy objective functions.Finally, two real distribution network planning projectsdealing with primary distribution network in the large city andsecondary network in the rural area are studied. <s> BIB002 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> Abstract A new planning model is developed for solving sizing, sitting and timing of distribution substation and defining the associated service area. The model incorporates a non-discrete function that accurately simulates the different cost components of distribution substations. Voltage, capacity, power conservation and radial-flow constraints are included into the developed model. The efficiency of the developed model is illustrated using a detailed numerical example. <s> BIB003 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> Important research effort has been devoted to the topic of optimal planning of distribution systems. However, in general it has been mostly referred to the design of the primary network, with very modest considerations to the effect of the secondary network in the planning and future operation of the complete grid. Relatively little attention has been paid to the optimization of the secondary grid and to its effect on the optimality of the design of the complete electrical system, although the investment and operation costs of the secondary grid represent an important portion of the total costs. Appropriate design procedures have been proposed separately for both the primary and the secondary grid; however, in general, both planning problems have been presented and treated as different-almost isolated-problems, setting aside with this approximation some important factors that couple both problems, such as the fact that they may share the right of way, use the same poles, etc., among other factors that strongly affect the calculation of the investment costs. The main purpose of this work is the development and initial testing of a model for the optimal planning of a distribution system that includes both the primary and the secondary grids, so that a single optimization problem is stated for the design of the integral primary-secondary distribution system that overcomes these simplifications. The mathematical model incorporates the variables that define both the primary as well as the secondary planning problems and consists of a mixed integer-linear programming problem that may be solved by means of any suitable algorithm. Results are presented of the application of the proposed integral design procedure using conventional mixed integer-linear programming techniques to a real case of a residential primary-secondary distribution system consisting of 75 electrical nodes. <s> BIB004 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> We describe a multistage, stochastic, mixed-integer programming model for planning capacity expansion of production facilities. A scenario tree represents uncertainty in the model; a general mixed-integer program defines the operational submodel at each scenario-tree node, and capacity-expansion decisions link the stages. We apply “variable splitting” to two model variants, and solve those variants using Dantzig-Wolfe decomposition. The Dantzig-Wolfe master problem can have a much stronger linear programming relaxation than is possible without variable splitting, over 700% stronger in one case. The master problem solves easily and tends to yield integer solutions, obviating the need for a full branch-and-price solution procedure. For each scenario-tree node, the decomposition defines a subproblem that may be viewed as a single-period, deterministic, capacity-planning problem. An effective solution procedure results as long as the subproblems solve efficiently, and the procedure incorporates a good “duals stabilization method.” We present computational results for a model to plan the capacity expansion of an electricity distribution network in New Zealand, given uncertain future demand. The largest problem we solve to optimality has six stages and 243 scenarios, and corresponds to a deterministic equivalent with a quarter of a million binary variables. <s> BIB005 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> This paper is the first of a two-part paper on optimal distribution system planning. The horizon distribution planning problem and optimal distribution system model formulation are described. The horizon planning mission is to minimize future costs by determining optimal design parameters given assumptions about the future. Prior work addressed short-range and expansion planning of subsets or combinations of design parameters. The many distribution requirements and associated constraints inhibited an all-inclusive evaluation of total horizon design requirements for the 20+ year period. The proposed model and optimization formulation provides a generalized horizon planning approach and introduce a fully functioning comprehensive horizon planning model using a perspective that encompasses all necessary parameters and constraints. Parameters include: substation and distribution transformer capacities; number, size, and lengths of distribution feeders and secondary conductors; and primary voltage class. Optimal design voltage drops and reliability indices are determined. The horizon planning optimization application is described and solved in the second companion paper using continuous constrained nonlinear programming methods. The application is demonstrated with Snohomish PUD case studies <s> BIB006 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> This is the second part of a two-part paper in which application methods to enhance distribution horizon planning for a 20+ year period are described. The optimal horizon planning model encompasses all distribution design requirements for primary and secondary systems. The model formulation is described in the companion paper Part I. The model makes use of a generalized feeder layout viewed as a tree-like circuit serving a circular sector of a round or hexagonal service area of uniform load density. The horizon model optimization objective cost function and constraint equation applications are described in this paper. The horizon planning model is validated with analysis of case studies and sensitivity evaluations for input and output variables. The case studies are assessed under various conditions such as changing consumer density, consumer characteristics, cost of energy, and economic assumptions. The application of the model is demonstrated using Snohomish PUD future expansion assumptions with case studies to examine impacts of horizon assumptions on future conductor sizes, primary voltage classes, voltage drop designs, reliability, and consumer cost of service <s> BIB007 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> This study presents an ordinal optimisation (OO) method for specifying the locations and capacities of distributed generation (DG) such that a trade-off between loss minimisation and DG capacity maximisation is achieved. The OO approach consists of three main phases. First, the large search space of potential combinations of DG locations is represented by sampling a relatively small number of alternatives. Second, the objective function value for each of the sampled alternatives is evaluated using a crude but computationally efficient linear programming model. Third, the top-s alternatives from the crude model evaluation are simulated via an exact non-linear programming optimal power flow (OPF) programme to find the best DG locations and capacities. OO theory allows computing the size s of the selected subset such that it contains at least k designs from among the true top-g samples with a pre-specified alignment probability AP. This study discusses problem-specific approaches for sampling, crude model implementation and subset size selection. The approach is validated by comparing with recently published results of a hybrid genetic algorithm OPF applied to a 69-node distribution network operating under Ofgem (UK) financial incentives for distribution network operators. <s> BIB008 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> It is widely accepted that renewable energy sources are the key to a sustainable energy supply infrastructure since they are both inexhaustible and nonpolluting. A number of renewable energy technologies are now commercially available, the most notable being wind power, photovoltaic, solar thermal systems, biomass, and various forms of hydraulic power. In this paper, a methodology has been proposed for optimally allocating different types of renewable distributed generation (DG) units in the distribution system so as to minimize annual energy loss. The methodology is based on generating a probabilistic generation-load model that combines all possible operating conditions of the renewable DG units with their probabilities, hence accommodating this model in a deterministic planning problem. The planning problem is formulated as mixed integer nonlinear programming (MINLP), with an objective function for minimizing the system's annual energy losses. The constraints include the voltage limits, the feeders' capacity, the maximum penetration limit, and the discrete size of the available DG units. This proposed technique has been applied to a typical rural distribution system with different scenarios, including all possible combinations of the renewable DG units. The results show that a significant reduction in annual energy losses is achieved for all the proposed scenarios. <s> BIB009 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> The main goal of this paper is to present a Multistage Distribution network Expansion Planning (MDEP) problem in the presence of Distributed Generations (DGs) in a multi-objective optimization framework. The proposed model simultaneously optimizes two objectives: minimization of investment and operation costs and maximization of reliability index. The proposed optimization model is solved subject to AC power flow constraints to obtain the optimal configuration of feeders (adding and removing lines) including the optimal size of branch conductor, replacement of conductor for reserve feeders, and generated power of DGs. To include reliability concerns in the proposed MDEP problem, an analytical approach on the basis of graph theory is implemented to evaluate the Energy-Not-Supplied (ENS) index as an extra objective function. Also, in this paper, in order to identify Pareto optimal solutions of the multi-objective MDEP problem, a hybrid Particle Swarm Optimization (PSO) and Shuffled Frog Leaping (SFL) algorithm is implemented. A synthetic distribution test system is considered for the MDEP problem in a 4-year planning horizon. The results of the hybrid PSO and SFL algorithm are compared with those of the classical PSO and SFL methods. <s> BIB010 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> With the progressive exhaustion of fossil energy and the enhanced awareness of environmental protection, more attention is being paid to electric vehicles (EVs). Inappropriate siting and sizing of EV charging stations could have negative effects on the development of EVs, the layout of the city traffic network, and the convenience of EVs' drivers, and lead to an increase in network losses and a degradation in voltage profiles at some nodes. Given this background, the optimal sites of EV charging stations are first identified by a two-step screening method with environmental factors and service radius of EV charging stations considered. Then, a mathematical model for the optimal sizing of EV charging stations is developed with the minimization of total cost associated with EV charging stations to be planned as the objective function and solved by a modified primal-dual interior point algorithm (MPDIPA). Finally, simulation results of the IEEE 123-node test feeder have demonstrated that the developed model and method cannot only attain the reasonable planning scheme of EV charging stations, but also reduce the network loss and improve the voltage profile. <s> BIB011 </s> Overview of Electric Energy Distribution Networks Expansion Planning <s> B. MATHEMATICAL PROGRAMMING <s> Planning distribution systems without considering the operation status of multiple distributed generation (DG) units could result in constraining the network, lowering the utilization of its assets and minimizing the total DG capacity that can be accommodated. In this paper, the impact of multiple DG configurations on the potential of active network management (ANM) schemes is firstly investigated. Secondly, the paper proposes a multi-configuration multi-period optimal power flow (OPF)-based technique (MMOPF) for assessing the maximum DG capacity under ANM schemes considering 1) variability of demand and generation profiles (multi-period scenarios), and 2) different operational status of DG units (multi-configurations). The results show that the availability of DGs at certain locations could critically impact the amount of DG capacity at other locations. If DGs are properly allocated and sized at certain locations up to the optimal limits, even with a “fit-and-forget” approach, the total connected DG capacity can be maximized, with minimum utilization of ANM schemes. However, exceeding these optimal limits may lead to minimizing the total DG penetration in the long term, impacting the system reliability due to the operational status of multiple DG units, and consequently, imposing more investments on ANM schemes to increase the amount of connected DG capacity. <s> BIB012
If the decision variables are a mixture of continuous and integer variables, the optimization problem is called a mixed-integer programming (MIP) problem. Linear programming (LP) methods are dimensionally scalable, computationally tractable and robust methods BIB004 , where the optimality of the final solution can be mostly guaranteed. If the optimization problem has non-linear functions (as the problem constraints or the objective functions), the problem is known as non-linear programming (NLP). Such techniques are also utilized in 3DEP problems BIB006 , BIB007 , BIB012 , BIB003 . The MIP problems with non-linear objective functions or constraints, the problem is called mixed-integer non-linear programming (MINLP) BIB009 . Some references have employed an Ordinal Optimization (OO) method for solving the optimization problem with minimum computational burden BIB008 . Dynamic Programming (DP) is another mathematical method introduced in 1950 and copiously used to solve the multi-stage power system problems. DP does not need to linearize the objective function. However, this technique has a high computation complexity for solving the large-scale optimization problem BIB002 . Generally, DEP is a large-scale, complicated optimization problem in which the complexity will dramatically be increased with the size of the problem BIB010 and take long processing time. As a remedy, researchers proposed the decomposition methods to find the optimum solution within a finite number of iterations. To reduce the calculation time, J. F. Benders proposed Bender' decomposition in 1962  . The disadvantage of this technique though is the complexity of implementation BIB002 . Dantzig-Wolfe decomposition (DWD) is developed by George Dantzig and Philip Wolfe to solve the LP problems with specific features known as complicated constraint problem  . DWD generates a problem consisting of BIB005 : A master problem (MP) to make decisions on binary variables, e.g. capacity expansions and on a base operational scheduling. A series of Sub-problems (SP) based on mixed-integer single period column generation Besides, reference BIB001 introduced the primal-dual interior-point (PDIP) method. Also, a modified PDIP technique was proposed in BIB011 to decide on the size of electric vehicle charging.