*4.2. Analytical Methods*

Analytical methods are briefly addressed with the emphasis that their capabilities are limited and that advanced computer intelligence methods are better used in multi-objective optimization problems. In Reference [37], the authors show the possibility of analytical calculation of voltage drops and power losses instead of iterative load-flow calculation used to determine the node in radial or doubly fed feeder where DG will provide the lowest power losses. The analytical approach described in the mentioned paper observes two objectives; the location of the distributed generation to obtain the lowest power losses and to fulfil the target voltage values along the observed feeder. If the second objective is not satisfied, the location is changed near the original solution until both objectives are met. Power dispatched from DG authors limit with the injected current value that cannot exceed the power consumption from the DG location. The proposed method can be performed with the following load-distribution constraints: uniform load distribution in the feeder, centralized load distribution in the feeder load where the largest power consumers are in the middle of the feeder and loads are increasing towards the feeder. When solving the allocation problem in ring and interconnected distribution networks, authors use the admittance matrix and minimize the objective function (1) [37].

$$f\_j = \sum\_{i=1}^{j-1} R\_{1i}(j) |S\_{Li}|^2 + \sum\_{i=j+1}^{N} R\_{1i}(j) |S\_{Li}|^2, j = 2, \dots, N \tag{1}$$

For the location of distributed generation in the node *j*, the aim is to find the lowest value of losses which is in the function of the equivalent resistance between the first and the *i*-th node. Additionally, the authors have investigated the possibility of renewable energy production volatility representation through a series of different data. Since the authors did not use iterative procedures there is no problem of convergence of the proposed method.

Achary et al. [25] are considering the GA usage in order to optimize the location of distributed generation but are abandoning the idea due to many iterative load-flow calculations which significantly affected the calculation time consumption needed for optimization. In their work, they use the theorem of the complex power that represents the basis for determination of the most sensitive node where distributed generation achieve the least loss in the system. The proposed solution according to the expression (2) will result in the optimal distributed generation dispatch for each node, while the contribution of each distributed generation to total system losses will be determine using theorem of the complex power. The node "I" with the least losses represents the optimal location of the distributed generation considering the consumed power "PDi" at that node.

$$P\_{DGi} = P\_{Di} + \frac{1}{a\_{ii}} \left[ \beta\_{ii} Q\_i - \sum\_{\substack{j=1,\ j \neq i}}^{N} \left( a\_{ij} P\_j - \beta\_{ik} Q\_j \right) \right] \tag{2}$$

Gözel et al. [26] developed a similar method to the one developed by the authors of Reference [14], with significant changes in the representation of the results in relation to the authors of Reference [25]. A significant contribution to their work stems from the definition of mutual influence of distributed generation node to other nodes in the distribution network. The results of their calculations are also based on power losses calculation as decisive and clear indication of distributed generation influence. In addition, the authors compare their results with the authors of Reference [25] and concluded that their method consumes less computing time.

Aman et al. [38] presented the Power Stability Index (PSI) method in order to find nodes in network which have the most favorable impact on voltage profile and grid losses when distributed generation is connected to them. The authors tested their method on a 69-bus test system and a significant contribution is shown defining the new voltage stability indicator for valuation of the nodes. PSI method is compared to Golden Section Search Algorithm (GSSA) and results are compared with strong similarities, but GSSA used more computation time when compared to PSI-based method given by expression (3), in which *PG* represents active power of distributed generation, *rij* stands for real part of line impedance and *Vi* is the real voltage of the *i*-th bus with the voltage angle, *δ*.

$$\text{PSI} = \frac{4r\_{ij}(P\_L - P\_G)}{\left[|V\_i|\text{Cost}(\theta - \delta)\right]^2} \le 1\tag{3}$$

## *4.3. Computational Intelligence–Based Methods*

Significant development of CI and numerous papers have pointed the applicability of selected metaheuristic methods used as higher order procedures to determine sufficiently good solutions without the necessary knowledge of the entire mathematical model or the values of all variables. Solutions derived from metaheuristic approaches are result of search for global best solutions in the predesignated and finite search space [64,65]. If the search space consists of many possible solutions and solution variants, metaheuristic methods often repeat some of the properties and re-evaluate already visited space. That feature often results in more precise and refined solution. Classification of metaheuristic methods is given by Figure 3 which presents most viable algorithms, grouped by type, governance rules, modeling rules and the way they determine the best solution.

Computational intelligence implies hybrid approaches created by combining several optimization methods and is characterized by successful application for continuous values, the ability to self-evaluate and change the way of execution, stochastic approach, parallelism in execution and the ability to generate approximate or Pareto solutions [66]. Procedures and methods contained in computer intelligence often base their principles of operation on biological principles or natural processes [67] and are applied in solving problems for which there are no effective or specialized procedures [68]. The lack of specific procedures for specific problems may be caused by complexity that does not

allow effective modeling or the inability to explain or model certain problems and all observed factors [69]. In power engineering, computer intelligence can be used to solve many optimization challenges, calculate optimal power flows [70–72], system modeling and monitoring of ADNs using fuzzy logic systems with evolutionary algorithms and artificial neural networks, while some procedures and algorithms can be used in data and event analysis and diagnostics of ADNs using qualitative reasoning, planning methods and hybrid procedures [73]. In addition, computer intelligence is successfully used to solve problems of designing electromechanical oscillation stabilizers, determining the causes and sorting of faults in the transmission network, reliability assessment, consumption forecasting, power system protection coordination, electricity quality assessment, economic supply of electricity, reactive power optimization and determination of optimal power flows—basically everywhere where iterative processes need faster convergence [74].

An extensive review of optimization procedures used in mathematical modeling in 360 scientific papers was provided by Theo et al. [7] outlining the advantages and disadvantages of each approach. For a GA, the authors outline suitability for solving problems that may have more favorable solutions, it is generally easy to integrate into an existing simulation framework, it is tolerant to the objective functions with chaotic attributes, and it is suitable for topological and categorical variable optimizations. However, as a lack of a GA authors outline the convergence to the local best solution instead of the global best solution, long-term convergence and complex approach of determining the criteria for the optimization process termination. When reviewing the Particle Swarm intelligence algorithm, the authors outline the advantages of fast execution, flexibility and openness to the other soft computing procedures but warn that the algorithm requires

definition of coordinate motion system and a proper number of particles otherwise can result in local best solutions.

A comprehensive overview of optimization methods that can contribute to the more efficient integration of DG into the power system are given by Colmenar-Santos et al. [75] in their paper from 2016. The authors of Reference [75] state that the problem of the multi-objective optimization of various technology DG integration into a fully developed active network has not ye<sup>t</sup> been completely solved and outline the idea of the development of a robust distribution managemen<sup>t</sup> system called AMN (Active Management Network) whose role is a real-time operational managemen<sup>t</sup> of DG and other control devices in distribution networks. Analyzing the optimization methods and grouping seen in the literature, the authors of this paper represent the division of optimization approaches into three groups:


According to Reference [75], conventional approaches are analytical methods, power flow calculations, non-linear programming method and rule 2/3; approaches based on artificial or computer intelligence consider evolutionary algorithms, algorithm of simulated annealing, differential evolution algorithm, Particle Swarm algorithm, fuzzy logic systems, ant-colony algorithm, tabu search algorithm, artificial bee colony algorithm, firefly algorithm while hybrid approaches include methods of unification of the GA and fuzzy logic systems, GA and tabu search algorithm, GA and Particle Swarm Optimization (PSO) algorithm, GA and power flow calculation, PSO algorithm and power flow calculation, tabu search algorithm and fuzzy logic systems.

After reviewing all of the methods, the authors [75] state that solutions based on the swarm intelligence algorithm are complex in development if reliable global solutions are sought while fuzzy logic systems and hybrid systems are not sufficiently represented in the literature. The authors outline that premature convergence is extremely emphasized for the methods based on the evolutionary principles if solution approach is not detail enough, while other significant methods, such as the simulated annealing algorithm and ant-colony algorithm, have long execution time which excludes them for possible application in short-term planning.

The largest number of scientific papers where meta-heuristic optimization methods are applied for allocation of distributed generation use the GA method or the PSO method, and there are also papers that use a hybrid method as a compound of these two procedures as well. Singh and Goswami [39] use GA determined by the objective function (4) for solving specially designed multi-objective optimization problem, where the impact of the distributed generation integration on active power market price λ is considered beside the power losses reduction and voltage conditions improvement. The authors used the knowledge presented in Reference [76] when assuming distributed generation influence on the electricity price *Ci(DG)*, depending on consumer active power *PDi* and distributed generation active power *PDGi*.

$$P\_{\rm delent}^{\rm DG} = \sum\_{l=1}^{n} \left[ \left\{ \mathbf{C}\_{l}^{d}(\rm DG) \times \left( P\_{\rm Di} - P\_{\rm DG} \right) \times \boldsymbol{\Lambda} \boldsymbol{t} + \mathbf{C}\_{l}^{l}(\rm DG) \times \left( Q\_{\rm Di} - Q\_{\rm DG} \right) \times \boldsymbol{\Lambda} \boldsymbol{t} \right\} + \left\{ \mathbf{C}(\rm DG) \times P\_{\rm DG} \times \boldsymbol{\Lambda} \boldsymbol{t} \right\} + \boldsymbol{\lambda} \times P\_{\rm L}(\rm DG) \times \boldsymbol{\Lambda} \boldsymbol{t} \quad \text{(44)}$$

The presented considerations of the authors of Reference [39] successfully confirm the simulations on the radial distribution network model, where they prove the usefulness of the proposed method for the allocation of one or more distributed generation in radial distribution networks.

## 4.3.1. GA-Based Methods

GA is a space search heuristic that is inspired by biological process of natural selection in which the most potent individuals of one generation give birth to individuals of the new, improved, generation. Genetic sequences determine the possibilities of the individuals within the population. The individuals are rated based on fitness function and the most fit ones reproduce to design a novel individual with better set of genes. If modeled correctly, GA can bring real adaptive computer programs. Main genetic operators are crossover and mutation, with all its advantages and disadvantages and difficulties in coding physical and mathematical models from the power industry domain as genes [77]. Binary encoding can be used in discrete search areas, while for continuous values a real-value encoding or tree encoding scheme needs to be used [77,78]. The process by which binary GA operates, as described in Reference [79] is presented by Figure 4. The process starts with objective function formulation, as with any method, but after the initial population the GA operators take place. In this example, each individual is encoded with n number of genes and each gene consist of m number of bits, as given by Figure 4. The fitness evaluation is usually in power engineering done my means of co-simulation and calculations. GA operators are mandatory as they ensure that the algorithm observes the whole search space.

**Figure 4.** Binary GA general process diagram.

A fundamental feature of GA is competition among individuals and the principle of elitism. Elitism implies the survival of only those individuals who have a high fitness score, and all other individuals are rejected. When observing multiple objectives simultaneously, it is necessary to pair individuals that have an elitist grade according to one of the criteria, in order to obtain an offspring that can have elitist grades according to all criteria. In some cases, most often when multiple solutions have equal value, not numerically, but value to the observer, selection methods such as Tournament Selection and Rank Selection are used in which individuals compete among each other in random manner to prove the unquestionability of the best individual.

Injeti et al. [27] used simulated annealing method for optimization and compared results with results obtained with GA and PSO method. In Reference [27], the authors clearly stated constraints that should be considered when using advanced optimization methods of computational intelligence such as simulation limitations. Successful implementation of one method of computer intelligence and review of other methods used for allocation of distributed generation are clear indicators of the future research, as shown in the more recent paper of the same authors [80].

$$P\_{T,Loss} = \sum\_{i=0}^{n-1} P\_{Loss}(i, i+1) \tag{5}$$

$$P\_{\rm Loss}(i, i+1) = R\_{i, i+1} \cdot \frac{P\_i'^2 + Q\_i'^2}{\left|V\_i\right|^2} \tag{6}$$

In Equations (5) and (6) *PT*,*Loss* stands for total feeder losses and *PLoss* represents line losses, the main minimization objectives considered in Reference [27].

Abou El-Ela et al. [40] used a method based on the GA for DG allocation and examined the proposed method by linear programming. The same authors concluded that there is no significant deviation in the obtained results, thus confirming the usefulness of computational intelligence–based methods. The authors of Reference [40] define clear indicators with or without DG for voltage profile improvement—VPI according to (7); spinning reserve increasing—SRI according to (8); power flow reduction—PFR according to (9); and total line loss reduction—LLR according to (10) [40]. The developed-algorithm authors successfully tested on a model of a real distribution network in Egypt.

$$\text{Max VPI\%} = \frac{VP\_{\text{w/DG}} - VP\_{\text{wo}/DG}}{VP\_{\text{wo}/DG}} \times 100\tag{7}$$

$$\text{Max SRI\%} = \frac{SR\_{w/DG} - SR\_{wo/DG}}{SR\_{wo/DG}} \times 100\tag{8}$$

$$\text{Max PFR\%} = \frac{PF\_{\text{k,wo/DG}} - PF\_{\text{k,wo/DG}}}{PF\_{\text{k,wo/DG}}} \times 100\tag{9}$$

$$\text{Max LLR\%} = \frac{LL\_{\text{wo}/DG} - LL\_{\text{w}/DG}}{LL\_{\text{wo}/DG}} \times 100 \tag{10}$$

The benefits of DG are present with index MBDG—maximal composite benefits of DG—according to the authors of Reference [40]. For the limitations of the computer intelligence optimization process, the authors state the limitations are the maximum number of distributed generation units so in their paper authors analyze the distribution of one distributed generation in the distribution network and conclude that the nominal power of the power plant significantly affects the indicators determined by expressions (7)–(10) unlike the location—node—of the power plant. The authors present the energy balance of the observed system as a limitation of the optimization procedure and determine the condition according to which the power of total production in the observed network must be equal to the sum of all time-constant consumers and power losses. Similar authors use GA to allocate the remote measurement and monitoring units in the Smart Grid environment Reference [81], where they deeply explain the parameters of GA-based optimization process.

Biswas et al. [41] used genetic algorithm for the optimal DG allocation in order to reduce voltage sags in a real complex distribution network. Presented problems are perceived by the authors through the real power losses (RPL) reduction objective function (11) that takes into account quotient of product of voltage angle difference cosine and line resistance with voltage level *Aij*, and quotient of product of voltage angle difference sinus and line resistance with voltage level *Bij*; objective function for decrease of customers that can be affected by the voltage sags in observed time period (NF) (12) with the total load distributed, *SDIST*, and the load distributed for the *i*-th fault, *LDISTi* ; the objective function of DG integration costs reduction (13), which is determined by financial units per kilowatt of installed DG power (KC) and observed as the objective of determining the least power of DG which will meet the previous two objectives [41]. The constraints of the proposed algorithm that the authors have defined, similarly to previous authors, are load-flow constraints through the specific branches, voltage constraints and limitations of the number of DG units.

$$\text{RPL} = \sum\_{i=1}^{N\_b} \sum\_{j=1}^{N\_b} A\_{ij} (P\_i P\_j + Q\_i Q\_j) + B\_{ij} \left( Q\_i P\_j - P\_i Q\_j \right) \tag{11}$$

$$\text{Min } S\_{DIST} = \sum\_{i=1}^{N\_F} L\_{DIST\_i} \tag{12}$$

$$\text{Min }\mathbb{C}\_{DG} = K\_{\mathbb{C}} \sum\_{i=1}^{N\_{\mathbb{F}}} P\_{DG\_i} \tag{13}$$

The specifics of the proposed method based on the genetic algorithm is emphasized in the integrated use of power flow calculations when evaluating the benefit of the best solutions in each iteration. After evaluating the benefit of individual solutions based on the power flow calculation, an additional algorithm of voltage failure analysis is performed, which evaluates the impact of the proposed solutions on the number of time-constant consumers covered by voltage failures. This approach is innovative because it considers multiple objectives taking place as several separate optimization procedures. The way in which the authors graphically present the results of several iterations with a threedimensional surface is seen later, in other works.

The limitations of methods based on a genetic algorithm for solving more complex optimization problems are explained in the paper by Yang et al. [18] in which a hybrid method of two genetic algorithms is used with the aim of determining the maximum power of distributed generation, taking into account the voltage and technical limitations of the elements of the distribution network. The first genetic algorithm, determined by the objective function (14), determines the minimum power of distributed generation limited by the short-circuit power on the primary side of the competent transformer, voltage level and technical characteristics of the equipment in the system. The second genetic algorithm, according to the function of the goal (15), determines the power of distributed production that will meet the requirements of consumers of the observed derivative, while respecting the solutions of the first genetic algorithm.

$$Min\,f\_{1,i} = w\_1 \left(\sqrt{\sum\_{j=1}^{k} \left(P\_i\left(S\_{MVA\_j}\right)\right)^2}\right)^{-1} + w\_2 \left(\sqrt{\sum\_{j=1}^{k} \left(P\_i\left(S\_{MVA\_j}\right)\right)^2}\right) \tag{14}$$

$$\dim f\_{2,i} = w\_1 \left( P\_i \left( S\_{MVA\_j} \right) \right)^{-1} + w\_2 P\_i \left( S\_{MVA\_j} \right) \tag{15}$$

Using this hybrid approach and using two genetic algorithms, the authors determined the possible power intervals of distributed generation depending on the short-circuit power on the busbars of the parent network. Moreover, same authors showed that in some specific cases the usual meta-heuristic methods may give false solutions.

Methods based on genetic algorithms that solve optimization problems of reactive energy generation with the objective of reducing losses in the observed systems are particularly presented in papers [82,83] and the paper of the author <sup>L</sup>ópez-Lezama [84] where the optimal distribution of the DG is determined according to the electricity price criterion. In Reference [84], the advantage DG is presented through the possibility of achieving a higher generated power price from the market price due to the fact that DG is near to consumers and that feature results in fewer power losses. The same authors emphasize that repeated optimization procedures with the random initial settings resulted in almost identical solutions, thus confirming the usefulness of the proposed method.
