Recent advances on power distribution system planning: a state-of-the-art survey

Abstract

The rise in installation/reinforcement and maintenance costs of power distribution systems enforces the planner to adopt an optimal strategy during system planning. The planning should be uch that the designed system should economically and reliably take care of spatial and temporal load growth, and service area expansion in the planning horizon. Mathematically, this planning is a multivariable multi-objective optimization problem. During the past decade or so, several planning algorithms have been developed. The purpose of this survey is to present a comprehensive review of all the developments in the planning technology, which includes all the reported optimization models and solution strategies. This state-of-the art survey is systematically organized to serve as a stepping stone for future researchers as well as a planning guide for the engineers. The various planning models are grouped in a three-level classification structure starting with two broad categories, i.e., planning without and with reliability considerations. The tree grows further with subdivisions. The last level of this classification tree consists of different planning models and solution strategies. Comprehensive discussions on each planning model are presented.

1 Introduction

During the last decade, deregulation has resulted in significant restructuring of power systems. This motivates the planners to design efficient, reliable, and cost effective power networks. Thus, there is a great need for efficient distribution system planning algorithms. It is essentially an optimization process [1, 2] to obtain a number of design variables, such as: (i) size and location of the substation, (ii) number of feeders and their routes, (iii) number and locations of the sectionalizing switches, and (iv) radial or meshed network structure etc. The optimal values of these variables are determined by optimizing multiple objectives, such as: (i) minimization of the installation cost of new facilities (i.e., substations, feeders, and feeder branches), (ii) minimization of capacity up-gradation of the existing facilities, (iii) minimization of the operational (maintenance and energy loss) cost, and (iv) maximization of the system reliability. These objectives must meet network constraints, for example, substation and feeder capacity, and maximum allowable node voltage deviation etc.

Power distribution system planning can be of three types: (i) static planning, (ii) dynamic planning, and (iii) pseudo-dynamic planning. The static planning is a single step planning of new distribution network corresponding to a planning horizon (some years). In the dynamic planning, multiple steps are performed until the planning horizon. Different scenarios of loads and connection points are considered for each step and a global optimization is performed. The pseudo-dynamic planning approach corresponds to the same formulation and data, but the optimization is made sequentially for each step. This leads to a solution generally worse than the previous one. However, in this approach, the optimization is simpler than the dynamic planning because it basically repeats static planning for each step. In each planning approach, the load growth is modelled over some planning horizon. For example, in static planning, the load growth is modeled in the form of new load nodes in an area for which a new substation is to be built. In dynamic/pseudo-dynamic planning, it is modeled in the form of growth of existing load and/or addition of new load nodes to an existing network. The latter type is also known as expansion planning.

Over years, researchers around the globe have contributed significantly to distribution planning optimization problem with several mathematical models and solution strategies. The literature has grown tremendously. Hence, it is essential to consolidate this state-of-the-art technology periodically in the form of review articles to benefit the researchers, academicians, and engineers. The literature survey shows three different reviews on distribution system planning [3, 5]. Among them, the most recent one is also a decade old. During the past decade, several new models and solution strategies for distribution system planning have been developed. Thus, it is necessary to organize the present-day technology in the form of a guidance diagram to assist the planning engineers and researchers in exercising various design options with ease. The salient features of this review are:

• Categorization of all available planning models

• Highlighting the basis of problem formulation for each model

• Main contributions of the models along with solution techniques

• Quality assessment of different planning approaches and future directions for this technology.

In this review, a three-level tree-structure is developed to classify different approaches as shown in Fig. 1. The Level #1 classification is on the basis of planning models without and with reliability features. The classification proceeds to Level #2 with two distinct categories: planning without and with uncertainty modeling. The former approaches use classical techniques with uncertain data being derived from forecasting methodologies. The planning with uncertainty modeling, which models the intrinsic uncertainties of future load demand, discount rate, fault occurrence (in planning with reliability feature) etc., is the recent addition. The classification tree grows further to Level #3 with different mathematical formulations of objective functions and associated solution strategies.

Fig. 1

Classification of distribution system planning schemes as a tree-structure

The paper is organized as follows. Sections 2 and 3 describe the planning models without and with reliability features, respectively, along with various solution strategies. A comparative quality assessment of the planning strategies is given in Sect. 4. Section 5 concludes the paper.

2 Planning models without reliability features

In planning models without reliability features, the planning optimization is done by minimizing the installation costs of new facilities (substation/feeder/feeder branch), the cost of capacity upgrade of the existing facilities, the maintenance cost, and the cost of the energy loss, under operational constraints. There are two approaches: planning without and with uncertainty modeling. In planning with uncertainty modeling, the uncertainty of future load demand is considered at the design stage.

2.1 Planning without uncertainty modeling

In this approach, the uncertain data are derived from forecasting methodologies. For example, the future load demand is derived using a certain load growth rate. The components of planning cost are aggregated into a mono-objective function which is optimized to determine the optimal values of the decision variables. There are three types of models in this category depending on the nature of optimizing (decision) variables/objectives: (a) mixed-integer, (b) discrete, and (c) continuous models. In general, the discrete/integer variables in distribution system planning problem are decisions on the installation of new substations/feeders/feeder branches and decisions on replacement/capacity addition of existing substations/feeders/feeder branches. The continuous variable is the power flow in each branch of the distribution network. In mixed-integer model, the optimal values of both discrete and continuous variables are determined. In discrete model, only discrete type of decision variables are determined. These three models along with their solution strategies are briefly discussed below.

(A) Mixed-integer models: The mixed-integer model is mostly used. It comprises of a set of integer variables, mainly binary decision variables $\{1 (\text{ Yes}), 0 (\text{ No})\}$ and a set of continuous variables, i.e., power flows in all branches. The objective function (total installation and operational cost) is expressed as:

$$\begin{aligned} \min \left( {\sum \limits _{i\in S_{af} } {C_i^B y_i +\sum \limits _{j\in S_{br} } {C_j^C P_j^2 } } }\right) \end{aligned}$$

(1)

There are two types of mixed integer models depending on the modeling of the cost of energy loss (linear [6–17] or quadratic [18–30]). For linear model, the quadratic function of cost of energy loss is linearized by piecewise linearization. In addition, the objective function is sometimes modified incorporating additional features, for example, a harmonic distortion factor is multiplied with the cost of energy loss in [20] to consider the harmonic distortion in industrial load. An additional constraint is used in [15, 16] to keep total investment cost within the utility CAPEX (capital expenditure) budget. In [17], the model optimizes an integral primary and secondary distribution network.

Solution strategies: There are various solution approaches adopted for these models. They are briefly discussed below.

• (i) Linear mixed-integer programming (LMIP) [7, 8, 10–12, 14–17]: The LMIP is a two-step approach. In the first step, a linear programming is solved using simplex method treating all variables as continuous variables to determine some initial solutions. In the second step, successive searches are performed to obtain better solutions for the integer variables.

• (ii) Quadratic mixed-integer programming (QMIP) [18, 21]: In QMIP, firstly, a quadratic programming is solved treating all variables as continuous variables. Then, the values of integer variables are converted into integer values using some suitable approaches. In [21], the strategy consists of two steps, i.e., firstly, few chosen networks (trees) are obtained for an unconstrained problem in the neighborhood of minimum spanning network; subsequently, the best one among them is chosen satisfying all the constraints with minimum cost.

• (iii) Branch and bound [6, 9, 19]: This is another type of mixed-integer programming where the lower bounds of cost for all branches are calculated. If the cost of any branch is greater than the lower bound, it is not considered. Thus, the success of this method depends on efficient ‘lower bound’ computation. Some major bounding criteria are devised as: minimal increment cost to achieve the lowest cost of energy loss, shortest path customer assignment for lower branch installation cost etc.

• (iv) Benders’decomposition [20]: This technique partitions the mixed-integer model to a pure 0–1 ‘relaxed master problem’ and a quadratic ‘sub-problem’. By solving the master problem, first decisions are made regarding construction of new facilities (substations/feeders). Then, the quadratic sub-problem is solved to optimize power flow for minimizing the operational costs.

• (v) Heuristic combination optimization algorithm [13, 22, 30]: In [13], a heuristic combination optimization algorithm is used. Heuristics are used to update solutions in successive iterations so as to determine a feasible and optimal solution. In [22], the problem is decomposed into five phases: decision on new facilities, commissioning dates, cost optimization of facilities, application of radiality constraint, and power flow cost optimization. The initial three phases are solved by branch and bound technique, and the remaining phases are solved by Tabu search. A heuristic-based algorithm is used in [30]. In this work, firstly, all binary decision variables are relaxed and the relaxed problem is solved by using non-linear programming (NLP). Then, the heuristic-based algorithm is applied to the solution obtained from NLP so as to determine a feasible solution and feasible values of binary decision variables. Finally a heuristic-based local improvement phase is used to improve the solution.

• (vi) Genetic algorithm (GA) [23, 25]: In [23], GA is used in expansion planning for comparatively larger systems. GA works on several genetic operators, such as selection, crossover, mutation etc. In [24], a static planning of urban network is solved using GA, where heuristics-based crossover and mutation operators are used to minimize the generation of infeasible solutions and an elitist strategy ensures survival of the better solutions. A problem specific GA called genetic shortest path algorithm is devised in [25], where random chromosomes are generated and global and local search are performed with genetic operators and the shortest path algorithm, respectively.

• (vii) Ant colony search (ACS) [26]: ACS is a class of evolutionary algorithms based on the abilities of ants to find shortest path to their food sources. The search in the ACS-based distribution system planning algorithm is guided by a heuristic guide function and probabilistic transition rule.

• (viii) Simulated annealing (SA) [27]: SA is a meta-heuristic optimization technique based on analogy with annealing of solids. It is a neighborhood search technique guided by a probabilistic criterion.

• (ix) Expert system [28]: The knowledge based expert system consisting of a data base for load data, location of nodes (and obstacles), and a set of heuristic rules is used. An inference machine is designed to obtain an optimal distribution network using the knowledge base.

• (x) Particle swarm optimization (PSO) [29]: PSO is another class of evolutionary algorithm that mimics the social behavior of a flock of birds and a fish school etc. In PSO, a population of particles searches for the optimal solutions using their individual own experience and by using guidance from their respective guide particles. A modified cost-biased encoding/decoding technique is devised in this work. The performance of different PSO variants is tested with statistical tests.

(B) Discrete models: In this model, the objective function formulation deals with discrete variables (binary), i.e., decision on provision of the network facilities. The discrete models reported in the literature are [31–41]:

(a) Simple discrete model [31–33, 35]: In this model, the optimal values of discrete decision variables, such as substation size, feeder routes are determined by minimizing the objective function:

$$\begin{aligned} \min \left( {\sum \limits _{i\in S_{af} } {C_i^B y_i } }\right) \end{aligned}$$

(2)

Solution strategies: The solution strategies adopted for these models are: (i) branch exchange [31, 32, 35], and (ii) multiple tasks based algorithm [33].

• (i) Branch exchange [31, 32, 35]: In single-stage branch exchange, a loop is created by adding a branch to the network and then a radial network is obtained by opening an optimally chosen branch. In [31], forward/backward path branch exchange is used, where a forward branch exchange, i.e., from year 1 to planning horizon year ($t_{a})$, is performed and then at each stage, backward branch exchange is done to regenerate some missing useful solutions. This process continues till $t_{a}$ is reached. In multistage branch exchange [32], the single-stage branch exchange algorithm is iteratively applied to obtain better solutions. In [35], a two-stage branch exchange is used, i.e., inter-zone, to find the optimal service area of each distribution substation zone and intra-zone, to optimize network under each distribution substation zone.

• (ii) Multiple tasks based algorithm [33]: The multiple tasks based algorithm proposed in [33] consists of a sequence of subroutines for different tasks, such as allocation of new feeders by extracting partial load of existing overloaded feeders by dual effective gradient method, determination of the least cost route from a new feeder to load nodes by the Dijkstra shortest path algorithm, finding the least cost route between load points and substation transformer by the Dijkstra method, determination of the minimum cost of the routes found by subroutines 2 and 3 by Ford–Fulkerson method, and finding the least cost tie-lines between a new feeder and the existing feeders by Tabu search.

(b) Multiple policies based discrete model [34]: The planning horizon is divided into few stages ($\eta _{sg} )$ in this model. A state in each stage consists of the existing facilities. A decision (e.g., $y_k^d$ in stage $k)$ is a plan for commissioning new facilities to meet load growth. A policy is defined as a set of decisions for all the stages. The policy cost is a recursive cost for all the decisions. The objective is to find out the optimum policy that minimizes total cost as given below.

$$\begin{aligned} \min \left\{ {C_k^p (y_{_k }^d )+\sum \limits _{k=1,...,\eta _{sg}} {C_{k+1}^p (y_{_{k+1} }^d )} } \right\} \end{aligned}$$

(3)

Solution strategy: Branch exchange is used in [34] to obtain the optimum policy solution in two steps: (i) static step: all decisions satisfying the constraints in each stage are determined; (ii) dynamic step: optimal policy is developed by minimizing the cost of decision in each stage using branch exchange.

(c) Reinforcement plan based discrete model [36]: In this model, the decisions are the discrete states consisting of different reinforcement plans, such as branch conductor replacement, building of new branches and substations etc. The state ($i$,$j$) represents the current plan $j$ in year $i$. State (0,0) is the existing facilities at the start. A movement from state ($i-1,j)$ to state $(i,m)$ represents the plan $m$ being realized in year $i$ from a previous plan $j$ subject to constraints. The transfer cost from state $(i-1,j)$ to state $(i,m)$ is represented as $C_{Tr} [(i-1,j),(i,m)]$ ($\eta _{st}$= number of states in each year). The objective function based on possible reinforcement plans in successive planning years is expressed as:

$$\begin{aligned} \sum \limits _{i=1,\ldots ,t_a } {\min _{m=1,\ldots ,\eta _{st} } \left\{ C_S (i-1,j)+C_{Tr} [(i-1,j),(i,m)]\right\} } \end{aligned}$$

(4)

Solution strategy: Dynamic programming is used in [36] to obtain the optimal plan among all viable reinforcement plans. A backward check is done to ensure non-repetition of plans. The optimal plans are determined using few heuristic rules: (i) conductor replacement for a feeder branch is done such that annual savings in energy losses equal the conductor replacement cost, and (ii) investment-optimized conductor replacement is done for more branches to eliminate voltage constraint violations.

(d) Scenario based discrete model: There are two models based on conductor selection and node location. They are:

• (i) Based on conductor-selection [37, 38]: This model is designed to sustain the system load growth by optimal conductor size selection (scenario) in successive planning years. Firstly, a set of allowable conductor sizes for each branch in each planning year is identified yielding a series of scenarios (conductor selections) for all branches meeting the constraints for the planning horizon. The branch conductor sizes to be used in successive planning years are the replacement scenarios, denoted as ‘scen’, within which the participation of a conductor size is represented as 1/0. The lowest cost scenario is named as ‘bestscen’ and the corresponding year of replacement is ‘bestnc’. This process is performed for all the branches. The objective function is defined as:

  $$\begin{aligned} \min \left( {\sum \limits _{j\in S_{sc} } {C_{Scen_j } scen_j } }\right) \end{aligned}$$

  (5)

  Solution strategy: Dynamic conductor planning used in [37, 38] to solve this scenario based model consists of the following steps: (1) start with an initial network for the first year, (2) optimize the network according to the scenarios bestscen, (3) perform possible modifications of the bestscen, (4) check for the network constraints and then realize the most economic conductor replacement scenario, and (5) repeat steps (2)–(4) for each successive year.

• (ii) Based on node location [39]: In this approach, a scenario represents a set of possible node locations and a policy for each planning stage is defined as the set of nodes to be connected with an existing network. The problem is dealt as a two-stage expansion problem with multiple scenarios with a common first stage policy. For example, in a two-stage and two-scenario case, if the first scenario is the node set {(a,b),(c,d)} and the other scenario may be {(a,b),(e,f)}. The first stage policy is to connect nodes (a,b) and second stage policy is to connect rest of the nodes. Mathematically, $s$ stands for a scenario of a planning stage and policy po is a mapping from scenario to a family of policy decisions, i.e., $po:s\rightarrow \{\partial ^s\}$ where, $\partial ^s=\{\partial _1^s ,\partial _2^s ,\ldots ,\partial _{t_a }^s \}$ ($\partial _j^s = \text{ decisions}$ for installation of new feeder branches in $j$th year, $t_a = \text{ total}$ planning years). The admissible policies (Ap) meet the constraints. The implementable policies (Ip) have same first stage decision component. The admissible as well as implementable policies are the only feasible policies (Fp). The objective function is formulated as:

  $$\begin{aligned} \min \left( {\sum \limits _{i\in Fp} {wt_i C_i^s } }\right), \quad \text{ where}\quad (Fp\in Ap\cap Ip) \end{aligned}$$

  (6)

  Although the title of this work contains the keyword ‘uncertainty’, the uncertainties in load demand and cost are not addressed in the problem formulation; hence, this model is grouped under planning without uncertainty modeling.

Solution strategy: GA is used to find the optimal set of policies in this model. The procedure is: (1) the problem is decomposed into a number of sub-problems (one for each scenario) and each scenario evolves through GA by generation of candidate solutions, (2) if any solution is not implementable, it is penalized, (3) each implementable scenario is evaluated by decomposing it into few subsets, evaluating the cost of each subset, and taking the dynamically-weighted sum of the subset costs, (4) candidate solutions evolve through GA, and (5) go for next generation till the termination criterion is reached.

(e) Multiple modules-based discrete model [40]: This model consists of four different modules. The switch status estimation module determines the digital information from overloaded feeders by load flow. This information is fed to an expert system based training set building module which determines the possible expansion plans. These plans are different discrete decisions for installation of new facilities. The artificial neural network (ANN) module generates an optimal plan. The graphic module contains the geographical information system (GIS) and helps in displaying the details of the optimal distribution network generated by ANN. The objective function is:

$$\begin{aligned} \min \left( {\sum \limits _{j\in S_{EP} } {C_j^{EP} y_j } }\right) \end{aligned}$$

(7)

Solution strategy: ANN based on back-propagation learning rules in association with switch status information module, training set module, and graphic module is used to generate optimum distribution network. The ANN is trained to generate some unique solutions capable of diverting the load from the overloaded feeders. The optimum solution is chosen that requires minimum objective function.

(f) Multiple construction-based discrete model [41]: In this model, different construction plans are taken based on power demand and voltage/current capacity constraints. If the power demand exceeds substation capacity, decision on installation of new facilities (tie-lines, section switches, feeders) is taken. For current/voltage constraint violations, decision to exchange the feeder branch with thicker conductors and installation of new facilities are taken. The objective is to determine the optimal construction plan minimizing the objective function given below.

$$\begin{aligned} \min \left( {\sum \limits _{j\in S_{CP} } {C_j^{CP} y_j } }\right) \end{aligned}$$

(8)

Solution strategy: Reactive tabu search is used in this work to determine the optimal construction plan in each year. During this process, firstly all possible and necessary construction plans are generated in each year and all possible paths of network transitions within a planning horizon are determined. Among these paths, the most cost effective one is chosen as the final solution.

(C) Continuous models: In [42, 43], a nonlinear continuous model is provided for multi-year planning of the distribution systems. This model has the following objective function:

$$\begin{aligned}&\min \left(\sum \limits _{i\in S_s } {C_i^{ss} \left\{ 1-e^{-wt_i \left(P_i /P_{\max _i } \right)}\right\} } \right. \nonumber \\&\qquad \quad \,\left.+\sum \limits _{i,j\in S_{br} } {C_{i,j}^{fb} \left\{ 1-e^{-wt_{i,j} \left(P_{i,j} /P_{\max _{i,j} } \right)}\right\} }+\sum \limits _{j\in S_{br} } {C_j^C P_j^2 } \right) \end{aligned}$$

(9)

If a potential substation $i$ (feeder branch $i$,$j)$ is utilized during planning, its $P_{i} (P_{i,j})\!>\!0$; so the exponential terms tend to zero leaving only its installation cost. The installation cost of an existing facility is considered as zero. The model only has continuous decision variables, i.e., power flow in feeder branches/substations. Thus, a large number of discrete decision variables in the optimization problem is avoided. A spatial load forecasting is performed in [43] to determine the load demand in each planning year.

Solution strategy: The continuous model is solved using dynamic planning [42, 43]. The steps of this approach are: (1) determine an initial set of permissible expansion activities, (2) determine load demand for each node for a given intermediate year, (3) solve the nonlinear objective function to determine the optimal network structure, and (4) repeat steps (2)–(4) for all planning years.

2.2 Planning with uncertainty modeling

A fuzzy quadratic mixed-integer model is developed in [44] to incorporate the uncertainties of future load demand and investment costs into the planning model. The uncertain data are modeled as possibilistic quantities (triangular fuzzy numbers) using fuzzy set theory. Thus, the objective function also becomes a possibilistic quantity and the fuzzy $\alpha$-removal approach is used to determine its equivalent crisp quantity. The objective is the minimization of the fuzzy removal of total installation and operational cost as given below.

$$\begin{aligned} \min \left\{ {\text{ Rem}\left( {\sum \limits _{i\in S_{af} } {\tilde{C}_i^B y_i +\sum \limits _{j\in S_{br} } {\tilde{C}_j^C \tilde{P}_j^2 } } }\right)} \right\} \end{aligned}$$

(10)

The level of $\alpha$ is a measure of risk for the planner and $(1-\alpha )$ is a measure of planning robustness.

Solution strategy: Tabu search is used to solve this model. A neighborhood search is carried out to update the solution till a termination criterion is satisfied.

3 Planning models with reliability features

It is a challenge for power system engineers to supply uninterrupted power economically [45]. Thus, the reliability considerations are incorporated into the planning models [46–82]. There are two ways to incorporate reliability features into the planning: (i) planning under normal conditions, and (ii) planning under contingency conditions. In the first approach, an additional objective function called reliability objective function is included in the model to maximize network reliability. Typical reliability objective functions are expected outage cost, expected annual non-delivered energy etc. In the second approach, a network is planned under some predefined fault/contingency conditions.

3.1 Planning under normal conditions

The planning under normal conditions is performed in two ways: (i) planning without uncertainty modeling, i.e., using experience-based or historical data, and (ii) planning with uncertainty modeling, in which uncertainty of faults, future load demand, cost etc. are modeled either by possibilistic or probabilistic approaches.

3.1.1 Planning without uncertainty modeling

There are two planning models in this category, i.e., (i) aggregated mono-objective model and (ii) multi-objective model. In the first approach, the total installation and operational cost are aggregated with the reliability objective function to obtain a single best solution. In the multi-objective model, cost and reliability are treated separately because of their conflicting nature. They are simultaneously optimized using the Pareto-dominance principle [83].

3.1.1.1 Aggregated mono-objective models

The three planning models under this category are discussed below.

(A) Mixed integer models:

This model deals with both continuous and discrete variables. Brief notes on different models follow.

(a) Feeder outage-based model [46–49]: In this model, the total installation and operational cost is modeled as in the quadratic mixed-integer model. The feeder outage cost is used to maximize network reliability. In [46], an aggregated outage cost attributed to control equipment, distribution transformer, and feeder branches is modeled as the reliability objective function. In [47, 48], the reliability objective is modeled on the basis of feeder outage cost (excluding outage costs of transformer, control equipments etc.). In general, the overall objective function is:

$$\begin{aligned} \min \left( {\sum \limits _{i\in S_{af} } {C_i^B y_i +\sum \limits _{j\in S_{br} } {C_j^C P_j^2 } } +\sum \limits _{j\in S_{br} } {C_j^{out} \lambda _j d_j l_j P_j } +\sum \limits _{i\in S_{Ceq} } {C_i^{out} \lambda _i d_i P_i } }\right)\qquad \end{aligned}$$

(11)

In [49], the reliability cost is formulated with load shedding cost which includes feeder outage cost along with the outage cost of associated protection and switching devices.

Solution strategies: A number of solution strategies are used to solve this model. They are:

• (i) Mixed-integer programming [46]: A sequential optimization based on mixed-integer programming is performed starting with substation location (and numbers) optimization followed by feeder routes and outage cost optimization.

• (ii) GIS enhanced GA [47]: This is used to solve the feeder outage-based model in [47]. The data is extracted by a GIS based data bass. The shortest path to connect different substations is found by network best path analysisof GIS module and the optimal network is obtained by using GA.

• (iii) SA [48]: The steepest descent method is used to obtain an initial candidate solution and thereafter it is further improved by SA.

• (iv) Branch and bound [49]: In this approach, a sequential optimization is carried out. Firstly, a pool of feasible solutions is determined using branch and bound algorithm. Then, the solutions are evaluated with many reliability indices, for example system average interruption frequency index (SAIFI), system average interruption duration index (SAIDI), average system availability index (ASAI), customer interruption duration (CID) at each load node, Customer interruption frequency (CIF) at each load node and expected energy not supply (EENS) to select a final solution.

(b) Optimal switch placement-based model [50]: In this model, a reliability objective (RB) is modeled as ‘expected annual non-delivered energy’ by placing sectionalizing switches. The reliability objective is formulated as function of:$RB_U$, i.e., upper bound of RB assuming no switches in the network (disconnections occur only at substations; under a fault condition, total isolation takes place yielding maximum non-delivered energy), $RB_L$, i.e., lower bound of RB with assumption of all branches being equipped with switches, and $\varepsilon _k \in [0,1]$ i.e., an ‘improvement coefficient’ to simulate the effect of a compromise solution in switch location policy. The overall objective function is:

$$\begin{aligned} \min \left\{ {\sum \limits _{i\in S_{af} } {C_i^B y_i +\sum \limits _{j\in S_{br} } {C_j^C P_j^2 } } +\varepsilon _k RB_U +(1-\varepsilon _k )RB_L } \right\} \end{aligned}$$

(12)

The total installation and operational cost is modeled as in the quadratic mixed-integer model. In [51], a combined reliability objective function is used based on optimal switch placement and feeder outage.

Solution strategies: The solution techniques used for solving models in this category are given below.

• (i) GA [50]: In optimal switch placement based model, GA is used to obtain the optimal network. In GA, a population of solutions evolves till convergence. In the process, dc load flow is performed to filter out infeasible solutions and ac load flow is performed to evaluate the solution based on the objective function.

• (ii) Network flow programming [51]: In this strategy, a distribution system is considered as a directed network graph. The entire network is assumed to have one source node (substation) and multiple sink nodes, connected by artificial arcs with maximum capacities (total cost of arcs is the substation cost). The optimal arcs are found by the shortest path technique. The flow-procedure starts with zero initial flow in each arc and flow amount is gradually increased till a balance is reached. During this procedure, reliability objective is optimized by minimizing the total reliability cost of all arcs, and locations of switches are optimized by placing switches, by trial-and-error, from the farthest point to the closest point of the substation.

(B) Discrete models: The objective function deals with discrete variables. Two discrete models have been used. They are:

(a) Discrete-state model [52]: The model decomposes the planning horizon (year) into few stages. Each state consists of a set of discrete state variables. The number of source transformers, their ratings, and conductor sizes of feeder branches are considered as discrete state variables at a particular stage, characterized by the objective function to reach that state. The system load growth in successive stages is handled by combinations of these variables that constitute some pre-determined states in each stage. This model incorporates reliability features on the basis of feeder branch and substation’s source transformer outage cost and the resulting cost of non-delivered energy. The objective function is:

$$\begin{aligned}&\min \left\{ \sum \limits _{i\in S_{af} } {C_i^B y_i +} \sum \limits _{i\in S_{st} } {\left( {C_i^{out} \lambda _i P_i +C_i^{nde} \lambda _i d_i P_i }\right)} \right.\nonumber \\&\quad \qquad \, \left.+\sum \limits _{j\in S_{br} } {\left( {C_j^{out} \lambda _j l_j P_j +C_j^{nde} \lambda _j d_j l_j P_j }\right)} \right\} \end{aligned}$$

(13)

Solution strategy: Dynamic programming is used as the solution strategy in this work, in which total cost to reach a new state from various preceding states is calculated and the transition associated with the minimum cost is stored. This analysis is performed for all possible new states. The state associated with the minimum cost at the end of the planning period is the optimum solution.

(b) Multiple policy-based discrete model [53]: In this model, firstly nodes are extracted from a GIS grid and distributed in different stages, i.e., stage-0 contains only substation and all other nodes are arranged in different stages according to their geographical locations. A state in a stage is defined as subset of nodes which can feed all nodes from the current stage to the final stage (stage-$M)$. The decision $y_m^d$ is defined as a path to move from one state of stage $m$ to another state of stage $m+1$. A policy is defined as the set of decisions from the current stage to the final stage, i.e., ($y_m^d ,y_{m+1}^d ,\ldots ,y_M^d )$. The decisions are based on the Markovian principle, i.e., any future policy stage is independent of the past policies. The objective function is modeled as policy cost $C_m^p$, i.e., a recursive cost function from stage$m$ to the final stage ($M)$. The reliability objective function is modeled as the cost of non-delivered energy. The objective function is the total policy cost which is sum of total installation and operational cost and the cost of non-delivered energy. The total policy cost in a stage $m$ is:

$$\begin{aligned} \min \left( {\sum \limits _{i=m}^M {C_i^p } y_i^d }\right) \end{aligned}$$

(14)

The overall objective is the minimization of the total policy cost of all stages.

Solution strategy: A back-propagation type dynamic programming is used to find the optimal policy in this work. The algorithm is: (1) set load locations using GIS, (2) determine stages 0 to $M$, (3) find the possible states for all the stages, (4) determine the decision for every stage, (5) determine the optimal policy for final stage using minimum cost criteria, (6) repeat the procedure for stage $M-1$ to 0, and (7) stop at optimal policy for stage 0.

(C) Continuous models: In [54, 55], a nonlinear continuous model is used for primary and secondary distribution network planning. A primary network is divided into several nearly-circular service areas of uniform load centered on the substations (Fig. 2) with radial feeders and lateral branches being perpendicular to the main feeder. A secondary network consists of distribution transformers and secondary branches. The model optimizes the total installation and operational cost and reliability objective function to determine the optimal values of decision variables such as number and length of main feeder per substation, substation capacity, primary voltage level, conductor sizes, number of secondary transformers and secondary lines per distribution transformer, and number of consumers per secondary line. All variables are continuous variables during optimization and, after optimization, the discrete variables are obtained by rounding-off operation. The reliability feature is incorporated using reliability indices, i.e., SAIFI and SAIDI. SAIFI (SAIDI) is computed as number (duration) of interruptions per consumer of secondary lines. The objective function is formulated as:

$$\begin{aligned} \min \left( {\sum \limits _{i\in S_{af} } {C_i^B y_i +\sum \limits _{j\in S_{br} } {C_j^C P_j^2 } } +C^{\mathrm{int}}.SAIFI.N_{C-sub} +C^{dur}.SAIDI.N_{C-sub} }\right) \nonumber \\ \end{aligned}$$

(15)

In [56], a similar continuous model is used with reliability being modeled as an expected outage cost.

Fig. 2

Planning of primary distribution systems: a typical substation area topology, b a typical substation service area

Solution strategies: This type of continuous models is solved using the following solution techniques.

• (i) Nonlinear programming (NLP) [54, 55]: NLP based on Kuhn–Tucker (KT) equations, as necessary optimality conditions, is used. Convergence to the global solution occurs after several test runs starting from some random choices for the optimizing variables. The integer decision variables are evaluated by rounding off final values.

• (ii) SA [56]: In this approach, the objective function consists of an aggregated cost and reliability objectives and it is solved using SA.

 

3.1.1.2 Multi-objective models

In this model, simultaneous optimization of cost and reliability is performed. The two types of multi-objective models are discussed below.

(A) Mixed-integer models [29, 57–63]: This model formulates objective function-1 as total installation and operational cost and objective function-2 as reliability cost (summation of outage cost and cost of non-delivered energy). The simultaneous optimization of these objectives is done using the Pareto-dominance principle [83]; a solution dominates the other if it is better in at least one objective without being worse in the other objectives. The solutions which are not dominated by any other solution are named as non-dominated solutions. This model provides a set of non-dominated solutions representing different network structures. It is to be noted that a part of the work reported in [29] is such type of multi-objective planning. In general, the two objective functions are:

$$\begin{aligned}&\min \left( {\sum \limits _{i\in S_{af} } {C_i^B y_i +\sum \limits _{j\in S_{br} } {C_j^C P_j^2 } } }\right)\end{aligned}$$

(16)

$$\begin{aligned}&\min \left\{ {\sum \limits _{j\in S_{br} } {\left( {C_j^{Ot} \lambda _j l_j P_j +C_j^{NDE} \lambda _j d_j l_j P_j }\right)} } \right\} \end{aligned}$$

(17)

Solution strategies: A brief discussion on the various solution strategies used is given below.

• (i) Benders’ decomposition [57]: In this approach, the mixed-integer planning problem is decomposed into two sub-problems, i.e., (a) linear programming problem to determine optimal values of continuous variables, and (b) integer programming problem to determine optimal values of the integer variables. The non-dominated solutions are generated using the $\varepsilon$-constraint method where any one objective function is optimized by specifying bounds to each remaining objective function. The specified bound for the $k$th objective is represented by $\varepsilon$ $_{k}$. Multiple non-dominated solutions are obtained choosing different values of $\varepsilon$.

• (ii) GA [58–62]: In [58], in potential solutions in an iteration are ranked according to objective functions and the non-dominated solutions are stored in an archive. The GA-based evolution stops when a desired number of non-dominated solutions are obtained. A solution with the total installation and operational cost more than an allowable limit is discarded. In [59], the Non-dominated Sorting GA-II (NSGA-II) [83] with elitist strategy is used. Elitism inserts the best solution directly in next iteration. The non-dominated solutions are obtained by evaluating fitness and the crowding-distance-assignment [83] to diversify the search. The potential solutions for next iteration are obtained by stochastic tournament [83] and problem specific crossover and mutation operators. In [60], two well known multi-objective GA, i.e., NSGA (Non-dominated Sorting GA) and SPEA (Strength Pareto Evolutionary Algorithm) [83] are used. SPEA2 (Strength Pareto Evolutionary Algorithm-2), an improved version of the SPEA, is used in [61]. The edge-set encoding technique is used along with some special recombination and mutation operators to obtain feasible solutions. In [62], a multi-objective GA is proposed based on network dataset built by component-GIS. An individual solution in the population is ranked based on number of solutions dominating the solution. This ranking and the elitist strategy assist the search to obtain non-dominated solutions.

• (iii) PSO [29, 63]: Multi-objective PSO (MOPSO) is used as solution strategy. The selection of guides is very much crucial on the performance of MOPSO. This is done using a well established multi-objective algorithm, i.e., SPEA2. A variant of PSO, i.e., comprehensive learning PSO, is used in [63] along with SPEA2 for guide selection. The performances of the algorithms are assessed with statistical tests.

(B) Discrete model [64]: This model performs trade-off between total installation and operational cost and network reliability by a rule-based expert system. The reliability is modeled using the indices SAIFI and SAIDI. Load growth is handled by some load reallocation plans to reallocate some load from overloaded feeders/substations to new feeders/substations. The optimization variables are discrete load reallocation plans based on total installation and operational cost satisfying reliability indices.

Solution strategy: The expert system used in solving this model consists of inference engine and knowledge base. The knowledge base is supported by heuristics rules on load reallocation. The inference engine is supported by minimum cost algorithm, minimum loss algorithm, and reliability indices computation algorithm.

3.1.2 Planning with uncertainty modeling

The uncertainties of future load demand, costs, and feeder branch faults are modeled using two approaches, i.e., possibilistic (fuzzy based) approach and probabilistic approach. The possibilistic approaches model the ambiguity of an event and the probabilistic approaches model the likelihood of occurrence of the event. It models the uncertainty associated with failure/future loads by fuzzy set theory and the optimization is performed in fuzzy domain. On the contrary, a probabilistic model handles the uncertainty by deriving the uncertain parameters from a chosen probability distribution function. The existing models are discussed below.

(a) Possibilistic model [65–69]: In [65–69], a possibilistic model is reported with fuzzy total installation and operational cost and fuzzy non-delivered energy as objective functions. The first objective is a function of fuzzy cost coefficient, fuzzy power flow etc. The second objective is a function of fuzzy failure rates and repair durations of feeder branches. The fuzzy $\alpha$-removal technique is used to find their equivalent crisp values. A third objective function, named as exposure index, measuring the risk of constraint violations is also included. The exposure associated with a feeder branch/substation (node) is the minimum $\alpha$-level at which the fuzzy power flow (voltage) is lower than power capacity (voltage) limit. The level of $\alpha$ is a measure of risk for the planner and ($1-\alpha$) is a measure of planning robustness. The final value of exposure index of a solution is obtained by aggregating the exposures of the capacity limit constraints of all branches and substations, and the exposures of the voltage limit constraints of all nodes. In [68], a GIS enhanced long term planning is provided. In [69], a planning robustness index, similar to the exposure index is used. It is computed by aggregating the robustness indices of all constraints, i.e., cable and transformer thermal limits, node voltage deviation limit, and maximum allowable SAIFI and SAIDI. In general, the two objective functions for this model are:

$$\begin{aligned}&\min \left\{ {\text{ Rem}\left( {\sum \limits _{i\in N_T } {\tilde{C}_i^B y_i +\sum \limits _{j\in N_b } {\tilde{C}_j^C \tilde{P}_j^2 } } }\right)} \right\} \end{aligned}$$

(18)

$$\begin{aligned}&\min \left( {\sum \limits _{j\in N_b } {\tilde{\lambda }_j \tilde{d}_j l_j \tilde{P}_j}}\right) \end{aligned}$$

(19)

Solution strategies: Brief discussions on different solution strategies for possibilistic model follow.

• (i) Fuzzy mathematical programming [65]: An interactive approach for multi-objective optimization in fuzzy environment is provided. In interactive approach, a decision maker (DM) guides the search according to its preference. Two fuzzy numbers are compared using some special indices, namely optimistic and pessimistic indices which reflect how far apart or close two fuzzy numbers are. These indices can be controlled by adjusting some safety parameters. The algorithmic steps are: (1) determine an initial solution for some given values of safety parameters assigned by the DM, (2) optimize each objective at a time using $\varepsilon$-constraint method to obtain a set of non-dominated solutions, (3) submit the set to the DM to select the best solution, and go to the next step if it is found to be satisfactory, else go to step (2), and (4) verify the solution for constraint violation and repeat the steps (1)–(4) if there is any constraint violation; otherwise stop the procedure.

• (ii) Tabu search [66, 67]: This is a local search technique in the neighborhood of non-dominated solutions. The search is performed by random inclusion/exclusion of new facilities into the network. The multi-objective optimization is done by partitioning the objective space into several search spaces for improvement of objectives. The non-dominated solutions are stored and updated before satisfying stopping criteria. In [67], the existing algorithm is improved by a new multi-objective Tabu search.

• (iii) GA [68, 69]: In [68], GIS enhanced CADDiN software is used to solve the problem with following steps: (1) define actual data and location of nodes and analyze the relevant data, (2) find all possible new branches to connect new nodes considering future load demand (by spatial load forecasting based on fuzzy reasoning) (3) calculate the costs of all possible new branches with available conductor sizes, and (4) use GA to optimize the network topology and conductor sizes. In [69], the problem is solved by a master GA and a slave GA. Firstly, the master GA finds out all possible new nodes, substations, and feeder branches required for final planning year and then, optimizes the connections with the existing system. The master GA calls the slave GA in each iteration to optimize the optimal timing (planning year) of commissioning each new element.

(b) Probabilistic scenario-based model [70]: In this approach, the uncertainties of future load demand and energy tax for each node (variable net income per unit energy delivered) are modeled. The values of these quantities are determined using a Gaussian probability distribution function. Firstly, all possible network structures are determined using artificial immune system algorithm. Then, a set of scenarios with different loads and energy taxes at all nodes are generated by Monte Carlo simulation. Each network is evaluated for all scenarios by four objective functions, i.e., total installation and operational cost, infeasibility rate (i.e., ratio of number of scenarios in which the solution violates constraints to total number of scenarios), mean cost of feasible scenarios (mean cost of the networks in the feasible scenarios), and the cost of non-delivered energy using the Pareto-dominance principle to obtain a set of non-dominated solutions.

Solution strategy: The probabilistic scenario-based model is solved by clonal selection algorithm for distribution networks (CSA-DN) based on artificial immune system (AIS). The CSA-DN determines all feasible networks (branches encoded as variables). It uses a mutation operator to diversify the search space by random insertion/removal of a branch. Among all the solution networks, multi-objective sensitivity analysis is done to obtain a set of non-dominated solutions.

3.2 Planning under contingency conditions

In this category, distribution networks are planned using some predefined contingency conditions so that the networks are capable of supplying a large area with faults at some specific locations. The two types of models in this category are discussed.

3.2.1 Planning without uncertainty modeling

The objective functions and the constraints in this planning model are formulated using experience-based data, such as load demand. There are two planning models in this approach: (i) aggregated mono-objective model and (ii) multi-objective model. They are described below.

3.2.1.1 Aggregated mono-objective model

In this model, cost and reliability objectives are aggregated. Different such models are given below.

(A) Mixed-integer models [71–73]: The optimal feeder routing, installation times for long term planning, and load reallocations among substations during some pre-determined contingencies are determined. A different continuous variable $L_{j,k}^R$ (amount of load transfer from substation $j$to $k)$ is used in the model to obtain the amount of load to be transferred during contingency conditions. The facilities involved in load reallocation incur additional cost. The objective function for this model is:

$$\begin{aligned} \min \left( {\sum \limits _{i\in S_{af} } {C_i^B y_i +} \sum \limits _{j\in S_s ,k\in S_k^{LR} } {C_{j,k}^{LR} } L_{j,k}^R }\right) \end{aligned}$$

(20)

The cost of energy loss is excluded in [71, 72]. In [73], the optimal network is found by minimizing the total installation and operational cost and customer interruption cost. Then, the optimal operational status of different facilities, for example, tie switch, capacitor bank etc. is determined to obtain optimal restoration of the network under different contingency conditions.

Solution strategies: The solution techniques used for solving this type of model are discussed below:

• (i) Mixed-integer programming [71, 72]: The branch and bound integer programming module of a software package is used for solving the model.

• (ii) GA [73]: A three-stage optimization using GA is used. It minimizes the total installation and operational cost in the first stage and determines optimal allocations, capacity selection, and replacement alternatives of system devices in the second stage. In the third stage, the optimal restoration problem is solved to find the optimal operational coordination of the system devices in contingency conditions. The initial population for GA is created using problem specific heuristics.

(B) Discrete models [74–77]: In this model, an expansion plan includes excess/reserve facilities (branches) to maintain supply in contingency. The objective function is similar to the simple discrete model (mentioned above). But, it is optimized under some predefined fault conditions. The system configuration for each fault case is determined by decomposing the problem into several sub-problems. A sub-problem is solved by single facility installation followed by multiple facilities installations (if required) to reduce constraint violations. The number of facilities to be installed per fault is predefined and is problem dependent. After solving all the sub-problems, the decomposition-coordination technique is used to optimize the cost by removing some of the unnecessary facilities or by changing some costlier facilities by cheaper ones. The discrete decisions are binary decision variables for installation of new facilities. In [75], an additional factor incorporated is contingency analysis by specifying predefined fault cases in different parts of the network. The model optimizes the network to reduce out-of-service area (in the fault cases). A multiple construction based model to determine the optimal construction plans under contingency conditions is provided in [76]. The objective function is similar to Eq. (8) and it is optimized under predefined contingency conditions.

Solution strategies: The following solutions strategies are used in solving this type of model.

• (i) Branch Exchange [74, 75]: The algorithmic steps are : (1) create an initial tree structure in first year, (2) perform branch exchange for each fault case, (3) reduce the sum of constraint violations of all fault cases by single facility expansion followed by multi-facility expansions, (4) minimize the sum of constraint violations using all candidate facilities, (5) reduce installation cost by removing unnecessary facilities or by replacing costly facilities with cheaper ones, (6) repeat steps (2)–(5) until best solution is found after coordinating all sub-problems for all the fault cases, and (7) repeat steps (2)–(6) are for all the years in planning horizon. In [75], multi-stage branch exchange is done by forward/backward path branch exchange.

• (ii) Expert system-based Tabu search [76]: The multiple construction based model is solved by Tabu search with a contingency analysis performed by a hybrid algorithm of expert system and Tabu search. It is a two-step process. A subroutine is used in the main algorithm to minimize the out-of-service areas under a fault. Firstly, the expert system performs multistage switching operations to find on/off status of the switches (i.e., tie-switch/section switch) during contingencies to shift some load from a feeder to neighboring/new feeders. Then, the Tabu search determines the optimal combination of the switching status based on which the optimal switch locations are decided.

3.2.1.2 Multi-objective model

In [77–79], a contingency-based reliability objective, i.e., Contingency-Load-Loss Index (CLLI), is formulated. It is a ratio of the average non-delivered load due to failure of all branches, considered one at a time, to the total load. It does not require the information on failure rates and repair durations of feeder branches. CLLI is mathematically expressed as:

$$\begin{aligned} CLLI=\frac{NDL_{avg} }{L_{total} }=\frac{\left( {\sum \nolimits _{i=1}^{\eta _b } {NDL_i }}\right)\big /\eta _b }{L_{total} } \end{aligned}$$

(21)

The multi-objective planning has been carried out for both radial and meshed networks using CLLI and total installation and operational cost in [77, 78]. The number, sites, and sizes of DG units are optimized in [77] using a second stage multi-objective optimization. In [79], a two-stage planning approach is reported. The optimal locations of sectionalizing switches along with optimal number of feeders and their routes are determined in the first stage of planning and optimal number and locations of tie-lines are determined in the second stage of planning. The two objective functions used in this work are also the total installation and operational cost and the CLLI. The optimal number of shunt capacitor banks is determined in addition to the feeder routes in [80]. The objective functions formulated in this approach are total installation and operational cost and a network performance metric comprising of node voltage deviation, power loss, and the CLLI.

Solution strategy: Multi-objective PSO (MOPSO) is used as solution strategy in [77–80]. A heuristic-based guide selection for MOPSO (HSG-MOPSO) is reported and its performance is shown to be better than SPEA2-MOPSO. The sensitivity of each parameter of the HSG-MOPSO is shown with statistical tests.

3.2.2 Planning with uncertainty modeling

In this planning approach [81, 82], the objective functions are formulated by considering load demand uncertainty so as to determine the optimal number of feeders, their routes, and optimal number and locations of sectionalizing switches. The load demand uncertainty is formulated using fuzzy (possibility) set theory. In this approach, a multi-objective model is formulated, in which the fuzzy objective functions are defuzzified using fuzzy removal technique so as to optimize them. The two objective functions are fuzzy removal of total installation and operational cost and risk factor comprising of fuzzy removal of CLLI and risks of node voltage and branch current limit deviations. In [81], only radial networks are considered. However, both radial and meshed networks are considered in [82]. In addition, the advantage of networks obtained the planning with fuzzy load in view of DG integration is brought out in [82].

Solution strategy: The HSG-MOPSO is also used as the solution strategy in both works [81, 82]. Several case studies are used to show the efficacy of the planning with load demand uncertainty over the planning with deterministic load demand. It is illustrated that the networks obtained with planning under load demand uncertainty are much more capable of sustaining overloading and load variations. The networks obtained with planning under load demand uncertainty are more efficient in DG integration as shown with case studies in [82].

4 Qualitative assessment of different planning models and solution strategies

The distribution system planning problem is a complex optimization problem due to the involvement of the following factors: (i) different types of decision variables (discrete/continuous), (ii) simultaneous choice of optimum conductor size and feeder routing, (iii) objective function consisting of linear as well as nonlinear terms, (iv) uncertainty of future load demand, (v) simultaneous optimization of the conflicting objectives, i.e., cost and reliability, (vi) considerations of geographical obstacles and social issues regarding site selections (substation and feeder routes), and (vii) highly constrained optimization search space. Very few works have incorporated all these issues in the planning model; most studies are based on simplified optimization models with several assumptions. The mixed integer model is the most generalized and widely used model consisting of both discrete and continuous variables. However, most studies avoid nonlinear optimization and use linearization of the nonlinear terms. Thus, the accuracy is sacrificed. The discrete model (reinforcement planning or multiple policy based model) has unique advantage in long-term planning as they can determine the actual timings of reinforcement plans and policies. This model also suffers from the same difficulty for large systems as in the mixed-integer model, since it determines the optimal plans from a large number of plans under the constraints. In continuous models, all the variables are considered as continuous variables eliminating the need of many discrete decision variables. This model is suitable for large systems. But, the difficulty lies in the round-off operation of the values of the discrete variables ensuring global optimum.

In planning without uncertainty modeling, all future data used in the planning model are derived from forecasting methodologies. Thus, the solutions obtained with this approach may not be robust enough to work in any future operating conditions (say, load demand). The planning with uncertainty modeling models the uncertainty of the future data using either possibilistic approach or probabilistic approach. As the probabilistic approach relies on the measurement of data, for example, mean values and extreme limits of the deviations from the mean values (errors), the success is very much dependent on the availability of actual data samples, knowledge of distribution of errors etc. On the contrary, in possibilistic approach, the ambiguity of uncertain data can be modeled efficiently without any problem specific knowledge. Another important issue is the incorporation of the reliability features into the planning model. Most of the works fall into the planning under normal conditions. The reason may be that the planning under contingency conditions requires some predefined contingency conditions which might be difficult to set beforehand. In planning under normal conditions, the aggregated mono- and multi-objective models are used. Since the network reliability and investment cost are the conflicting objectives, the multi-objective model which provides a set of non-dominated solutions is the most practical approach.

Amongst all solution strategies, it is found that the deterministic algorithms, i.e., linear/nonlinear mixed integer programming has better convergence properties than the heuristics based algorithms; but they suffer from the curse of dimensionality and difficulty in handling nonlinearity and non-convexity of the problem. Hence, smaller test systems are purposefully chosen in these works which use deterministic algorithms. For multi-objective planning approaches, the population based heuristic algorithms, such as evolutionary algorithms are very much useful because of the multi-point search capability, which helps to obtain a set of non-dominated solutions in a single run. Moreover, they can efficiently handle the nonlinear, non-convex objective functions. Thus, the applicability of the evolutionary algorithms is not limited to small scale systems, unlike the deterministic algorithms. Hence, the present research is more focused on the multi-objective evolutionary algorithms (MOEA). Some notable salient features of different planning approaches are summarized in Table 1. The most recent trend in this area of the research is towards incorporating DG in the planning stage. Hence, some planning approaches reported in the recent years are briefly described in the following subsection. Some possible future research directions in this area are also provided in another subsection.

Table 1 Some salient features of different planning approaches

4.1 The most recent trend: distribution system planning with DG

The most recent trend is the incorporation of DG directly in distribution system planning problem. A number of works [84–93] have been reported in the literature. A glimpse of all these works is given below:

• All the reported works are aimed at determination of optimal number, site, and size of DG units along with other network parameters. This is done by optimizing the total installation and operational cost with DG units. In some models, reliability cost is additionally incorporated into the objective function in the form of expected cost/value of non-distributed energy [89, 92, 93], customer outage cost [90] etc.

• Different solution strategies used are: mixed-integer programming [84, 87], branch and bound [85, 86], GA [89–91, 93], hybrid Tribe-PSO and ordinal optimization [92] etc.

• Some model formulation is done on deregulation environment, such as [84, 87] etc. In [84], the model considers the distribution company (DISCO) as the owner of DG. The decisions on generating power from DG and/or purchasing power from the main grid through existing substation or inter-tie are also determined with different case studies. In [87], the model consists of two sequential optimizing modules so as to determine: (i) optimal period and location of resources and (ii) resource capacities and production/contracts of the DISCOs.

• In some models, the uncertainty in load demand and generation is modeled by different probabilistic scenarios, such as, [89, 92, 93] etc.

• A long-term multi-stage model aimed at stage by stage (i.e., year by year) expansion planning including DG is given in [85, 86]. Different load levels are also considered in this approach.

• A multi-criteria decision making approach based on fuzzy logic is reported in [88]. The desirable limits/criteria on network power loss, cable loading, and node voltage are modeled by fuzzy sets. The model also incorporates some qualitative parameters, such as DG installation security, physical space etc.

• A pseudo-dynamic approach of multi-stage planning is provided in [90]. Different types of loads are considered in the planning.

• A dynamic planning model is reported in [91]. The novelty of this approach is consideration of variation of customer load using load duration curve and load-dependant electricity price.

• In [92], different renewable DG resources, such as wind, solar photovoltaic, and biomass are considered in the planning. The reactive capability limits of these DG resources are modeled. In addition, system uncertainties such as load demand, wind speed, and solar radiation are probabilistically modeled.

4.2 Some future research directions

More investigations are needed in the following areas:

• A long term multi-stage and multi-objective planning approach

• A unified approach for planning of both primary and secondary distribution systems in multi-objective planning

• Consideration of unbalanced secondary network in the multi-objective planning model

 

5 Conclusions

In this state-of-art paper, recent advances in distribution system planning have been extensively reviewed and discussed. A significant contribution of this paper is the systematic classification of all the existing planning models. This classification chart consists of a four-level tree structure for the planning problem viewed from different approaches for modeling and solution strategies in the presence/absence of possible uncertainties. All the existing mathematical models are discussed briefly along with adopted solution strategies and, in every case, the main essence of the planning is highlighted. In the end, a comprehensive qualitative assessment of all the models has been presented along with relative merits and demerits. It is hoped that this survey will be a handy resource for power distribution system planning engineers to have a summary view of the existing techniques, which, in turn, should enable them in arriving at a decision on the strategy to adopt under some specific circumstances. Some possible future directions of further work are suggested to extend the existing techniques in a more realistic and precise manner.