Optimal Clean Energy Resource Allocation in Balanced and Unbalanced Operation of Sustainable Electrical Energy Distribution Networks

Abstract

Electric power is crucial for economic growth and the overall development of any country. The efficient planning of distribution system is necessary because all the consumers mainly rely on the distribution network to access the power. This paper focuses on addressing distribution system challenges and meeting consumers’ fundamental needs, such as achieving an improved voltage profile and minimizing costs within an environmentally sustainable framework. This work addressed the gap in the existing research by analysing the performance of both balanced and unbalanced systems within the same framework, specifically using the IEEE 33-bus and IEEE 118-bus test systems. Unlike prior studies that focused solely on either balanced or unbalanced systems, this work redistributed balanced loads into three-phase unequal unbalanced loads to create a more challenging unbalanced distribution network. The primary objective is to compare the effects of balanced and unbalanced loads on system the performances and to identify strategies for mitigating unbalanced load issues in each phase. Six optimization methods (PSO, TLBO, JAYA, SCA, RAO, and HBO) were employed to minimize losses, voltage variations, and other multi-objective function factors. Additionally, the study compared the cost of energy loss (CEL), emission factors, costs associated with distributed clean energy resources (DCER), and active and reactive power losses. Phase angle distortions due to unbalanced loads were also analysed. The results showed that among the optimization techniques tested (PSO, TLBO, JAYA, SCA, RAO, and HBO), the HBO method proved to be the most effective for the optimal allocation of distributed clean energy resources, yielding the lowest PF_MO values and favourable outcomes across the technical, economic, and environmental parameters.

1. Introduction

In a traditional power system, electricity is generated in central locations; therefore, maintaining the power quality in the distribution network is a challenge. This energy transmits unidirectionally from the source to load through the transmission line. Recently, the electrical power system shifted from a traditional system to a modern restructured system that includes many renewable and clean energy sources [1]. Very low voltage was found at rural distribution feeder stations [2]. In distribution systems, the active and reactive power characteristics depend on the load, and the constant power load model is insufficient in distribution systems. The actual loads depend on the voltage magnitude, so an accurate voltage-dependent load flow model is required. The optimal sizing and location of DG are determined by considering system performance based objective indices such as active and reactive power losses, reliability, sensitivity factor and voltage deviation index using different types of loads [3]. Reference [4] describes probabilistic load flow, including random branch uncertainties in node power injection. In [5], uncertainty parameters created an unbalanced three-phase distributed radial system. Network planning, service restoration, and distribution network configuration were analysed in the different unbalanced loads. In [6], all the phases of an unbalanced network were modelled independently in decoupled mode. The tree structure technique was performed with the help of a forward/backward process and analysed rapidly without modifying the numbering node and branch. In [7], the author determined the optimal size and load of variable renewable DG sources (like biomass and solar energy) while considering multi-objective indices like voltage deviation, economic factors, and total active power loss. In [8], APSO was used to optimize effective cost-cutting and improve economic factors. In [9], the integration of DG was achieved using Butterfly-PSO; the parameter probability accelerated and enhanced the sensitivity for a quick optimal solution compared to PSO. In [10,11], optimal sizing placement of renewable DG was performed under seasonal load variations, which included industrial, residential, constant, and mixed load schedule planning for 24 h. In [12], fuzzy approach load flow was applied to balanced and unbalanced systems considering the composite load model. In [13], the authors proposed a method to optimize the location of a wind generator and improve the overall sustainable average load factor, power loss, and voltage profile by considering the hourly load power model. In [14], MINLP was used to optimize the voltage regulation uncertainty problem in a radial distribution network. In [15], uncertainty problems were modelled by fuzzy logic. In [16], the stochastic approach was used to reduce the cost of DG by appropriately optimizing the size and location. In [17], a cost–benefit analysis was conducted using the EHO algorithm. The optimization was based on a group of elephant behaviours, which improved the voltage profile and active and reactive power losses while considering the essential cost factor. In [18], the optimal location of DG, as well as the capacity of DG, was achieved using bat optimization to minimize the voltage stability index and real and reactive power losses. The dragonfly algorithm [19] was used to optimize the optimal placement of DG, including the economic benefits. In [20], the optimal location and size were determined by including the voltage stability factor under load growth. The total cost of the system was influenced by the power factor, stability margin, and load growth. In [21], DG placement was used to improve nodal pricing and the congestion effect. DG reduced the loss and congestion in the distribution system and maximized the profit by decreasing nodal pricing using an improved artificial bee colony algorithm technique. The three phase enhanced IEEE 33 bus benchmark test system for distribution system analysis is presented in [22]. In [23], the exponential load model was implemented using PSO with consideration of a multi-objective index. In [24], hybrid microgrids were implemented, and the different utility cost tariffs were analysed. The integration of DG and capacitors affects the market price [25]. In [26], the distribution system’s active and reactive power losses were reduced by reconfiguring the network. In the reconfiguration, the IEEE 33-bus system was converted into a five-mesh loop. The overall reliability and stability of the system improved after the reconfiguration. In [27], the hybrid optimization technique was used to reconfigure the network for reactive power compensation. The nonlinear behaviours of the loads and devices created harmonics. This time-varying harmonic power flow was analysed in the distribution system [28]. In [29], the harmonics analysis was modelled in Python. A detailed study of the different types of distribution system harmonics models was explained in [30]. The harmonics distortion model [31] has been implemented in metal factory industries. In [32], a PSO algorithm was developed, which was based on swarm behaviours. In [33,34,35], the TLBO algorithm was described, which originated from student and teacher learning behaviours. In [36,37,38], Rao developed JAYA and RAO optimization algorithms. The SCA algorithm [39] was proposed on the basis of sine cosine function features. The modelling of three-phase unbalanced systems is explained in [40,41]. The three-phase load flow model, power balance, and impedance matrix with the star delta model were implemented in a practical unbalanced distribution system. A three-phase transformer model considering the core loss and imbalance phase shift is discussed in [42]. In [43], the author presents a novel and fast NR load flow method for a three-phase unbalanced distribution network incorporating the Jacobian matrix. The decoupled harmonics power flow model [44], considering nonlinear loads, was tested and verified with the IEEE-13 bus network. In [45], IEEE standard definitions were presented to measure power quantities in sinusoidal and non-sinusoidal cases. Taguchi’s multi-objective approach [46] was used on a small and extensive distribution system to analyse the harmonics and relative angle distortion. The integration of non-dispatchable and dispatchable DG was utilized in distributed energy systems to improve the flexibility of energy-storing batteries and electric vehicles [47,48,49]. The integration of distributed generation (DG) [50] into distribution networks is covered in this article.

The authors of [51] concentrated on creating robust dynamic operating envelopes. They suggested the idea of a dynamic operating envelope, which enables DERs to adjust and perform at their best under changing network conditions in the context of unbalanced power distribution systems. In [52], the authors addressed unbalanced distribution systems, including those using renewable energy sources, and addressed the issues in LV distribution networks. The authors of [53] investigated the critical issue of inherent imbalances in distribution networks, which can lead to voltage fluctuations, power outages, and a reduction in system reliability. Sophisticated control solutions handle the issues related to imbalances in distribution networks through the application of the distributed model predictive control (DMPC) model for voltage regulation using single- or three-phase DG [54]. The authors of [55] used stochastic optimization or scenario-based approaches to consider the fluctuations in EV charging patterns and the production of renewable energy. In [56], the initial voltage estimation method was used to calculate the three-phase power flow for an unbalanced RDN. It discusses how the imbalance affects losses, voltage profiles, and the overall system performance. In [57], a multi-objective voltage control method with the GA algorithm was used to found the benefits from the trade-off energy. In [58], the combination of PEVs and renewable energy sources was linked to an unbalanced microgrid distribution system, and a multi-objective optimization technique was used.

Lotfi et al. [59] explored the integration of demand response aggregators in a reconfigurable distribution system, which also included photovoltaic (PV) and storage units. This study stands out due to its objective function of minimizing the operation cost and energy loss in time-dependent demand response programs. In [60], a multi-objective energy management strategy for 95-node distribution grids was developed that integrated energy storage units alongside demand response programs. In [61], the authors investigated the optimal sizing of distributed generation (DG) units and shunt capacitors within a distribution system, taking various uncertainties into account. In [62], the authors proposed using meta-heuristic wild horse optimization to deploy distributed energy resources (DERs) for electric vehicles (EVs) in unbalanced distribution systems. This work provides a comprehensive analysis of the adverse effects of high EV penetration, which is a significant challenge. In [63], the authors determined the capacitor placement in unbalanced distribution systems by applying graph theory. The authors utilized graph theory concepts to model the distribution system and identify the optimal locations for capacitor placement. The honey badger optimization techniques to solve the complex problems is described in [64,65].

Aljohani et al. [66] presented a two-stage optimization technique for addressing the volt/var optimization (VVO) problem in unbalanced distribution networks with a large penetration of plug-in electric vehicles. Their approach consists of two stages: the first focuses on optimal decomposition, while the second includes MILP into the volt/var optimization problem. Their findings showed that the proposed strategy considerably improves voltage stability and minimizes power losses in the network, making it a feasible alternative for managing the complexity brought on by high levels of EV penetration. Girigoudar et al. [67] investigated the impact of different voltage imbalance measurements on distribution system optimization. Their research focused on imbalance metrics to improve the performance and reliability of distribution networks. Many renewable energy mitigation strategies have negative consequences. This study aimed to decrease voltage imbalances and losses. Mousavi et al. [68] provided a framework for distribution system operators (DSOs) to support distributed energy resource (DER) aggregators in market participation in unbalanced distribution networks. Barutcu et al. [69] studied how harmonics affect photovoltaic (PV) penetration levels in unbalanced distribution networks. They emphasized the importance of better harmonic management strategies. Zheng et al. [70] proposed a powerful deep learning-based network reconfiguration technique for three-phase unbalanced active distribution networks. Their method improves the adaptability and durability of network topologies, especially in the face of changing operational conditions, by leveraging deep learning’s predictive capabilities. Hashmi et al. [71] looked at strategies to activate flexible and curtailable resources in three-phase, unbalanced distribution networks. They proposed that strategically deploying such resources can significantly increase the flexibility and stability of these networks, especially during times of fluctuating demand. Rafi et al. [72] conducted a thorough examination of imbalance compensation methods in active distribution systems using renewable energy, with a particular emphasis on the usage of power electronic converters. For unbalanced distribution feeders, Guo et al. [73] proposed a cooperative method for voltage regulation and peak shaving. This technology allows feeders that are severely unbalanced to remain operational by lowering the peak loads and regulating the voltage. Mohamed et al. [74] maximized energy savings in balanced and unbalanced distribution power systems by redesigning the network and optimally deploying capacitors. They employed a hybrid metaheuristic technique to significantly improve system reliability and energy efficiency by improving these two critical characteristics simultaneously. Souheyla et al. [75] introduced the Golden Jackal Algorithm, a nature-inspired optimization approach, to determine the optimal size and position of DG in UDNs. Their approach outperformed standard optimization strategies for DG placement and sizing.

The suggested model manages the uncertainties and minimizes the voltage deviation index and operating cost. A detailed overview of balanced/unbalanced distribution network studies on the basis of the system, objective, methodology, and outcome is shown in

Table 1. Studies on balanced and unbalanced distribution network development, including their objectives, methodology, and outcomes.

Author(s)	Bus System	Objective	Methodology	Outcomes
Das, D. et al. [1]	Radial distribution network (RDN)	To solve load flow problems for RDNs.	The proposed method solely focuses on evaluating the voltage magnitudes using basic algebraic expressions.	RDN load flow solution is a simple and efficient method.
Bohre, A.K. et al. [3]	Standard balanced IEEE 33, IEEE 54, and IEEE 69 RDNs	To find optimal DG sizing and placement using load models and soft computing approaches.	Optimal solution for an MOF is obtained using GA and PSO.	Proposed method offers economic benefits, enhanced reliability, and minimized losses.
Hu, Z. and Wang, X. [4]	The 24-bus RDN	To solve load flow using the probabilistic load flow technique with branch outages taken into account.	Probabilistic load flow method.	Proposed load flow approach can perform faster speed calculations that consider branch outages.
Rani, K. et al. [8]	Practical balanced 94-bus system	Optimal size and placement of renewable DG with load variation.	Solved by APSO.	Voltage index, active power loss, and cost-economic factor improvements were suggested.
Kalesar, B.M. and Seifi, A.R. [12]	Balanced and unbalanced RDNs	Composite load model with fuzzy load flow.	Fuzzy load flow method.	The results showed that this fuzzy load flow method can be used in large-scale balanced and unbalanced distribution systems.
Nayak, M.R. et al. [13]	IEEE 37 unbalanced RDN	Optimal allocation of BESS energy with wind power penetrations.	Inherited Competitive Swarm Optimization.	Showed a suitable sustainable average charging method to charge the battery, reduce the loss, and enhance the voltage profile.
Daratha, N. et al. [14]	Modified IEEE 123 unbalanced RDN	To fix the voltage regulation issue in unbalanced RDNs.	Problem solved by mixed-integer nonlinear programming (MINLP) and Monte Carlo simulations to obtain optimal result.	The results show that, even in the presence of generation and load uncertainties, the magnitude and imbalance ratio of voltages will always remain within the specified limits.
Suresh, M.C.V. and Belwin, E.J. [19]	IEEE 15, 33, and 69 balanced RDNs	To improve voltage profile, regulation, losses, and stability, and minimize the cost.	Dragonfly algorithm used for minimization of objective function and the results were compared with those of the Elephant Herding Optimization Algorithm and evolutionary algorithms.	Results showed a reduction in losses and cost using proposed method.
Murty, V.V. and Kumar, A. [20]	IEEE 12, modified 12, IEEE 69 and IEEE 85 bus	To decide on the best DG placement based on the revised voltage stability index.	Proposed voltage stability index method.	The proposed methodology showed an enhancement of voltage stability of the system under load growth.
Dashtdar, M. et al. [21]	38-bus RDN	To determine DG placement and appropriate size based on reducing nodal pricing using a nonlinear load model.	Improved Artificial Bee Colony (IABC) algorithm for optimization.	Showed a reduction in nodal pricing and indices of loss.
Mtonga, T.P. et al. [26]	IEEE 33 bus and IEEE 69-bus balanced RDNs	Reconfiguration of network.	Sparrow search algorithm.	Reduced real power losses and enhanced the efficiency and performance.
Xie, X. and Sun, Y. [28]	IEEE-13, 123, and 8500 tests	To analyse the probabilistic and time-varying harmonics.	Developed method for evaluating harmonic characteristics in unbalanced residential distribution systems.	Evaluated the probabilistic harmonic emission level.
Teng, J.H. and Chang, C.Y. [43]	Unbalanced RDN	To develop a novel and fast three-phase load flow method.	Proposed a unique and fast three-phase load flow approach for imbalanced RDNs.	The approach enhanced the efficiency and speed of load flow calculations.
Milovanović, M. et al. [44]	IEEE 13 unbalanced RDN	To develop a power flow method for nonlinear loads.	Introduced a power flow approach for nonlinear loads in unbalanced three-phase distribution networks.	Showed improved accuracy in analysing systems with nonlinear load components.
Meena, N.K. et al. [46]	Distribution systems	To find optimal integration of DG into distribution systems.	Multi-objective Taguchi approach (MOTA).	Optimal integration of DG and improved system performance and reliability.
Singh, P. et al. [47]	IEEE 33-bus benchmark test system	To find optimal distributed energy resource (DER) mix in RDNs.	Monarch Butterfly Optimization (MBO) with multi-criteria decision-making (MCDM).	Reduced annual energy loss and increased voltage stability margin.
Adewum, O.B. et al. [50]	UK electrical system grid	To study distributed energy storage in RDNs.	ESS integration methodologies were used to examine the effect of distributed energy storage on power quality.	Reduced the peak energy demand, improved DG benefits and reduced expansion costs.
Liu, B. and Braslavsky, J.H. [51]	33-bus and 132-bus unbalanced RDNs	To analyse the operating statuses of customers and the controllability of reactive powers.	The three-phase optimal power flow problem with linear imbalance was solved using a non-convex technique based on a geometric construction.	Maximized the available capacity with new sources.
Pinthurat, W. et al. [52]	LV distribution networks	Integration of renewable energy in LV distribution networks.	Review and study of LV distribution networks under unbalanced conditions.	Investigated the EV charging issues under unbalanced conditions.
Zhang, D. et al. [53]	IEEE 33 unbalanced RDN	Optimal battery energy storage system allocation.	Proposed an optimal BESS allocation mechanism to increase RDN dependability and economics.	Showed optimal allocation strategies using BESS to improve system performance and reliability.
Jiao, W. et al. [54]	Unbalanced distribution network with 45 loads	To minimize active and reactive power loss and voltage variation.	Implemented distributed voltage control using DMPC.	DMPC controller achieved the goal with both single- and three-phase DG.
Vijayan, V. et al. [55]	IEEE 123 test node feeder	Efficient modular optimization scheme.	This study proposed a modular optimization scheme designed considering the uncertainties in Electric Vehicle (EV) and Photovoltaic (PV) penetrations.	Achieved minimal voltage regulation and reduced peak demand and energy loss.
Yang, N.C. et al. [56]	IEEE 13- and IEEE 4-bus systems	Power flow calculations for a three-phase system.	Initial voltage estimation.	Found the feasibility and effectiveness of the unbalanced test system.
Tapia-Tinoco, G. et al. [57]	Modified IEEE 13 unbalanced RDN	To provide a technique for controlling ESs in real-time applications.	Modelling of electric springs using a continuous genetic algorithm and multi-objective voltage control.	Achieved optimized power losses, voltage deviation, and voltage imbalances.
Zandrazavi, S.F. et al. [58]	Modified IEEE 34 unbalanced RDN	Stochastic multi-objective optimal energy management.	Introduced an approach for stochastic multi-objective optimization for efficient energy management in grid-connected imbalanced microgrids with renewable energy and plug-in electric vehicles.	Minimized the operating cost of the system.

. Based on the above-mentioned comprehensive review, it was found that the verification and analysis of balanced and unbalanced distribution system performance, including DCER or DG planning considering technical and environmental parameters, has not been properly addressed. Therefore, the presented work fills this gap in the research on balanced and unbalanced distribution networks, including the technical and environmental parameters.

The key objectives of the proposed work were as follows: (i) to analyse the performance and behaviour of a three-phase distribution network under balanced and unbalanced load conditions; (ii) to assess the various performance aspects of the distribution network like active power loss, reactive power loss, voltage profile, pollutant emission, cost of energy loss, DCER cost, and payback year under balanced and unbalanced conditions; (iii) to verify the behaviour of the three-phase balanced and unbalanced distribution networks in terms of maintaining the minimum maximum voltage magnitude and its phase angle variation under nonlinearly distributed loads; (iv) to propose 33-bus three-phase balanced and unbalanced standard benchmark function systems and validate the results using IEEE 118-bus three-phase balanced and unbalanced systems; (v) to compare the performance and behaviour of the proposed system under both balance and unbalanced conditions using five different optimization techniques. Many research articles presented a performance analysis using the balanced IEEE system, but we believe that practical radial systems must experience some unbalanced load distribution, so this work focused on minimization of the mentioned issues under the unbalancing effect of all three phases separately and a comparison of the effects.

This paper is organized into seven main sections. The Section 1 provides an introduction, followed by the Section 2, which covers distributed clean energy resources and distribution networks. The Section 3 details the modelling of distribution networks. The Section 4 presents the formulation of the multi-objective performance function. The different soft computing techniques implemented in the proposed work are described in the Section 5 and the Section 6 is the results and discussion. Finally, the Section 7 concludes the paper.

2. Distributed Clean Energy Resources and Distribution Networks

Distributed clean energy resources are small-scale generation units connected to a distribution system that provide electricity to a distribution network. These includes equipment for extracting energy from renewable clean energy sources like solar panels, wind turbines, hydro turbines, hydrogen, geothermal and biogas energy resources. They can be used in various settings, including residential, commercial, and industrial facilities, and can provide backup power during power outages. Distributed generation involves an onsite energy generation and storage unit connected with the distribution grid. Distributed energy resources generally include renewable sources and improve the system’s load ability. DCER can be utilize as compensation to improve power quality problems. By adding DCER, many power quality issues can be solved, but the size, location, and cost factor are challenges for design. The power loss index, voltage distortion index, constant constraints, and economic and environmental factors are examined before installing modern DCER. Distribution generation is recognized as an advanced business model. Market players generate economic benefits through the installation of modern, efficient DCER.

2.1. Distributed Clean Energy Resources (DCER)

Clean energy resources produce minimal or no harmful pollutants or emissions that contribute to climate change. DCER reduce carbon emissions, improving the distribution network’s optimal resource allocation, uncertainty management, and sustainability [49,76,77,78,79,80,81]. Some of the most common outcomes and impacts of clean energy sources are represented in

Table 2. Different clean energy sources.

Clean Energy Source	Author(s)	Impact
Photovoltaic Energy	Dincer, F. [76]	Provide alternatives for policymakers and reduce emissions.
Solar Energy	Kumar, V. et al. [77]	Provide alternatives for coastal/offshore projects by utilizing a clean energy source.
Photovoltaic and Wind Energy	Fathi, R. et al. [78]	Reduce greenhouse gas emissions and allow for optimal clean energy resource allocation in distribution networks.
Hydrogen Energy	Tarhan, C. and Mehmet, A.Ç. [79]	Offer reduced emissions and provide clean and sustainable energy for the future.
Geothermal and Alternative Clean Energy	Ismail, B.I. [80]	Offer clean energy with reduced environmental emissions.
Biogas Energy	Surendra, K.C. et al. [81]	It improves sustainable energy usage in developing countries and promotes the utilization of clean energy resources.

.

2.2. Three-Phase Distribution Networks

In the radial distribution networks the R/X ratio is high because resistance variation is higher as compare to reactance. In radial distribution, a small-diameter conductor is used, which affects the R/X ratio. In radial networks, the distributed load characteristics are uncertain. The unbalanced distribution of loads creates a nonlinear behaviour in the system and affects the power quality. Protection schemes, unbalanced loads, transformer connections, and sudden load fluctuations are considered parameters in designing radial systems.

The three-phase enhanced IEEE 33 benchmark test system [22] includes five switchable branches and a transformer configuration in a radial distribution system. In this study, the switchable branches were removed, and the modified data are tabulated in Appendix A. The modified case system was connected to 3.715 MW of active power and 2.3 MVAr reactive power loads in balanced and unbalanced cases. In the balanced case, the load data were modified to have an equal distribution in all phases, but in the unbalanced case, the total load was distributed according to the enhanced IEEE 33 benchmark test system [22] without changing the three-phase line data. Figure 1a,b show the diagram for the balanced and unbalanced three-phase load distributions, respectively. A, B, and C are representations of the phases in a three-phase load. In this study, bus 1 was considered a slack bus.

3. Modelling of Distribution Network

A distribution system consists of a transmission line, transformer connection, shunt admittance, and load configuration. Soft computing is used to simulate its behaviour and performance under various conditions. The purpose of modelling is to analyse the system’s performance and integrate the distribution of clean energy resources to improve the voltage profile and reduce losses, emissions, and the effective cost of the system. These three-phase networks are represented in mathematical form to solve the load flow problem.

3.1. Modelling of Transmission Line

Three-phase balanced and unbalanced systems are conventionally modelled as symmetrical components. This study analysed the three-phase load flow with the help of the phase frame method [40]. The primitive impedance matrix is used to represent the mathematical model of the three-phase network impedances.

$$V_{br}^{abc}=Z_{br}\times I_{br}^{abc}$$

(1)

where $V_{br}^{abc}$ is the branch voltages, $Z_{br}$ is the primitive impedance matrix, and $I_{br}^{abc}$ is the branch current.

$$\left[\begin{array}{l}{{\displaystyle v}}_1^a\\ {{\displaystyle v}}_1^b\\ {{\displaystyle v}}_1^c\\ {{\displaystyle v}}_1^n\end{array}\right]-\left[\begin{array}{l}{{\displaystyle v}}_2^a\\ {{\displaystyle v}}_2^b\\ {{\displaystyle v}}_2^c\\ {{\displaystyle v}}_2^n\end{array}\right]=\left[\begin{array}{cccc}{{\displaystyle z}}_{aa}& {{\displaystyle z}}_{ab}& {{\displaystyle z}}_{ac}& {{\displaystyle z}}_{an}\\ {{\displaystyle z}}_{ba}& {{\displaystyle z}}_{bb}& {{\displaystyle z}}_{bc}& {{\displaystyle z}}_{bn}\\ {{\displaystyle z}}_{ca}& {{\displaystyle z}}_{cb}& {{\displaystyle z}}_{cc}& {{\displaystyle z}}_{cn}\\ {{\displaystyle z}}_{na}& {{\displaystyle z}}_{nb}& {{\displaystyle z}}_{nc}& {{\displaystyle z}}_{nn}\end{array}\right]\left[\begin{array}{c}{{\displaystyle i}}_{12}^a\\ {{\displaystyle i}}_{12}^b\\ {{\displaystyle i}}_{12}^c\\ {{\displaystyle i}}_{12}^n\end{array}\right]$$

(2)

$${{\displaystyle v}}_1^{abcn}-{{\displaystyle v}}_2^{abcn}={{\displaystyle z}}_{pr}^{abcn}{{\displaystyle i}}_{12}^{abcn}$$

(3)

where ${{\displaystyle z}}_{pr}^{abcn}$ is the primitive impedance matrix. In the impedance matrix, diagonal elements are self-impedance and non-diagonal elements represent mutual impedance.

The primitive admittance matrix is

$${{\displaystyle y}}_{pr}^{abcn}={\left({{\displaystyle z}}_{pr}^{abcn}\right)}^{-1}$$

(4)

where ${{\displaystyle y}}_{pr}^{abcn}$ is the three-phase primitive admittance matrix.

3.2. Modelling of Three-Phase Distribution Transformer

The three-phase distribution transformer was transfigured into a primitive admittance matrix [41]. This study considered a two-winding three-phase transformer, consisting of six coupling coils and three tertiary windings. The coupling coils and tertiary winding were ignored in the analysis. Figure 2a,b represent the three-phase star and delta impedance configurations.

$$y_{bus}^{abc}=c^T\times y_{pr}^{abc}\times c=\left[\begin{array}{cccccc}{{\displaystyle y}}_p& 0& 0& {{\displaystyle -y}}_{mu}& {{\displaystyle y}}_{mu}& 0\\ 0& {{\displaystyle y}}_p& 0& 0& {{\displaystyle -y}}_{mu}& {{\displaystyle y}}_{mu}\\ 0& 0& {{\displaystyle y}}_p& {{\displaystyle y}}_{mu}& 0& {{\displaystyle -y}}_{mu}\\ {{\displaystyle -y}}_{mu}& 0& {{\displaystyle y}}_{mu}& {{\displaystyle 2y}}_s& {{\displaystyle -y}}_s& {{\displaystyle -y}}_s\\ {{\displaystyle y}}_{mu}& {{\displaystyle -y}}_{mu}& 0& {{\displaystyle -y}}_s& {{\displaystyle 2y}}_s& {{\displaystyle -y}}_s\\ 0& {{\displaystyle y}}_{mu}& {{\displaystyle -y}}_{mu}& {{\displaystyle -y}}_s& {{\displaystyle -y}}_s& {{\displaystyle 2y}}_s\end{array}\right]$$

(5)

$$C=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& -1& 0\\ 0& 0& 0& 0& 1& -1\\ 0& 0& 0& -1& 0& 1\end{array}\right]$$

where $y_{bus}^{abc}$ and $y_{pr}^{abc}$ are three-phase admittance and primitive admittance matrices, and ${{\displaystyle y}}_p$ and ${{\displaystyle y}}_{mu}$ are self- and mutual admittance.

In this work, all primary and secondary windings, admittance and leakage admittance are converted in per unit values. All the transformer connections were configured as star winding in the three-phase balance case, as represented in Equation (5), and in the unbalanced case, both delta and star connection admittances were required for modelling. Equation (6) represents the delta-connected admittance matrix [41]. In the proposed model, leakage admittance was not considered in the analysis.

$${{\displaystyle y}}_{\Delta bus}^{abc}=\left[\begin{array}{cccccc}{{\displaystyle y}}_p& 0& 0& {{\displaystyle (-y}}_{mu}/\sqrt{3})& {{\displaystyle (y}}_{mu}/\sqrt{3})& 0\\ 0& {{\displaystyle y}}_p& 0& 0& {{\displaystyle (-y}}_{mu}/\sqrt{3})& {{\displaystyle (y}}_{mu}/\sqrt{3})\\ 0& 0& {{\displaystyle y}}_p& {{\displaystyle (y}}_{mu}/\sqrt{3})& 0& {{\displaystyle (-y}}_{mu}/\sqrt{3})\\ {{\displaystyle (-y}}_{mu}/\sqrt{3})& 0& {{\displaystyle (y}}_{mu}/\sqrt{3})& {{\displaystyle (2y}}_s/3)& {{\displaystyle (-y}}_{mu}/3)& {{\displaystyle (-y}}_{mu}/3)\\ {{\displaystyle (y}}_{mu}/\sqrt{3})& {{\displaystyle (-y}}_{mu}/\sqrt{3})& 0& {{\displaystyle (-y}}_{mu}/3)& {{\displaystyle (2y}}_s/3)& {{\displaystyle (-y}}_{mu}/3)\\ 0& {{\displaystyle (y}}_{mu}/\sqrt{3})& {{\displaystyle (-y}}_{mu}/\sqrt{3})& {{\displaystyle (-y}}_{mu}/3)& {{\displaystyle (-y}}_{mu}/3)& {{\displaystyle (2y}}_s/3)\end{array}\right]$$

(6)

where ${{\displaystyle y}}_p$, ${{\displaystyle y}}_s$, and ${{\displaystyle y}}_{mu}$ are the primary, secondary, and mutual admittance.

The three-phase branch current matrix can written as

$$\left[\begin{array}{c}{{\displaystyle i}}_p^{abc}\\ {{\displaystyle i}}_s^{abc}\end{array}\right]=\left[\begin{array}{cc}{{\displaystyle y}}_{pp}^{abc}& {{\displaystyle y}}_{ps}^{abc}\\ {{\displaystyle y}}_{sp}^{abc}& {{\displaystyle y}}_{ss}^{abc}\end{array}\right]\left[\begin{array}{c}{{\displaystyle v}}_p^{abc}\\ {{\displaystyle v}}_s^{abc}\end{array}\right]$$

(7)

where the ‘p’ suffix represents the primary winding and ‘s’ represents the secondary winding of the transformer in Equation (7).

3.3. Distribution Load Flow Methodology

The three phase backward/forward sweep load flow algorithm is used in the proposed work. This method computes all the bus voltages and branch currents, and then computes the active and reactive power losses and the parameter index. The backward forward load flow was based on KVL and KCL. The power and current flows are determined by backward sweep, while the bus voltages are calculated using the forward sweep method. The voltage was considered to be constant during the backward sweep, and the current and power were considered constant in the forward sweep [82].

The load flow algorithm is an iterative method that solves the power flow equations by sequentially updating the voltage magnitudes and angles at each bus in the system. In the backward sweep, the power injections at each bus are calculated using the load or generation data, and the magnitude and angle of the voltages at the load buses is updated. In the forward sweep, the slack bus is assumed as the reference bus, then the power flows of all transmission lines are calculated and finally, the bus voltage magnitude and angle towards the adjacent buses are updated. This methodology is used to analyse the performance of electrical power systems under steady-state conditions.

The nodal current injected at the nth node for the kth phase is

$$I_n^k={\displaystyle \frac{P_{n,k}+j{Q}_{n,k}}{V_{n,k}^{*}}}$$

(8)

where k denotes phases A, B, and C; $I_n^k$ is the node current; $P_{n,k}$ is the active power for each phase at node n; $Q_{n,k}$ is the reactive power for each phase at node n; and $V_{n,k}$ is the voltage at node n.

Backward sweep: In this stage, each branch current is updated starting from the end nodes of the distribution system.

$$I_{ij,k}={\displaystyle \sum _{n\in nodes}{I}_n^k}$$

(9)

where $I_{ij,k}$ is the branch current between the ith and jth nodes for the kth phase.

Forward sweep: In this stage, the nodal voltages are updated based on the updated branch current from the source node.

$$V_{j,k}=V_{i,k}-I_{ij,k}\times Z_{ij,kk}$$

(10)

where k represents all three phases (A, B and C) in the distribution network, $V_{j,k}$ _{is the} voltage at the jth node, and $V_{i,k}$ is the voltage at the ith node for the kth phase.

3.4. Constraint Considerations

Power system constraints are critical factors that must be considered in operation and planning to ensure safe, reliable, and efficient power delivery to consumers [83,84]. Power system design requires specific limits to sustain the stability of the system. In load flow power balance, voltage limits and active and reactive power demands must be specified within standard limits [85,86]. In the system, if any constraint values are outside of the limits, it can affect the stability of the system and damage the installed equipment.

3.4.1. Voltage Limit Constraints

Branch impedance and unbalanced loads affect the voltage profile in a three-phase system. The R/X ratio is typically higher in a practical distribution system, so the voltage should be maintained within limits to ensure system reliability.

So, the voltage reference limit is set as

$${{\displaystyle v}}_i^{\min }\le {{\displaystyle v}}_{bus}\le {{\displaystyle v}}_i^{\max }$$

Typically, it should be ±5% to ±10% of the nominal voltage according to the IEC 60038 [84] standards for distribution systems.

3.4.2. DCER Active and Reactive Power Constraints

The DCER rating must lie between minimum and maximum limits. Its capacity determines the cost and size of the system.

$$P_{DGi}^{\min }\le P_{DG}\le P_{DGi}^{\max }$$

$$Q_{DGi}^{\min }\le Q_{DG}\le Q_{DGi}^{\max }$$

The amount of active power that DCER can generate is limited by their capacity, and the quantity of reactive power depends on whether it is supplied or absorbed by DCER. In this study, the maximum values of DCER active and reactive powers considered are 1.1350 MW and 0.85 MVAr, respectively.

3.5. Performance Parameters

Distribution network performance depends on transmission line parameters like resistance, inductance, and capacitance.

3.5.1. Active and Reactive Power Losses

Active power loss is the power loss due to the resistance of the transmission and distribution lines, while reactive power loss is due to the reactive component of the power. Both types of power loss reduced the efficiency and increase the costs of the distribution network. The three-phase active power loss (${\mathrm{Ploss}}_{3-phase}$) and reactive power loss (${\mathrm{Qloss}}_{3-phase}$) were calculated using Equations (11) and (12).

$${\mathrm{Ploss}}_{3-phase}={\displaystyle \sum _{br=1}^{N_{br}}(|i_{br-a-ph}{|}^2}\times R_a)+{\displaystyle \sum _{br=1}^{N_{br}}(|i_{br-b-ph}{|}^2}\times R_b)+{\displaystyle \sum _{br=1}^{N_{br}}(|i_{br-c-ph}{|}^2}\times R_c)$$

(11)

$${\mathrm{Qloss}}_{3-phase}={\displaystyle \sum _{br=1}^{N_{br}}(|i_{br-A-ph}{|}^2}\times X_a)+{\displaystyle \sum _{br=1}^{N_{br}}(|i_{br-B-ph}{|}^2}\times X_b)+{\displaystyle \sum _{br=1}^{N_{br}}(|i_{br-C-ph}{|}^2}\times X_c)$$

(12)

where $i_{br-a-ph},i_{br-b-ph},i_{br-c-ph}$ are the branch currents and $R_a,R_b,R_c$ are the resistances, and $X_a,X_b,X_c$ are the reactances of the three-phase network.

3.5.2. Pollutant Emissions

The power system affects ecological systems, creating harmful pollutant emissions. These pollutants (CO, NO_x, SO₂, CO₂, etc.) negatively affect the human body, create health issues, and affect the respiratory system. The pollutant emission factor for CO₂ is 632.0 g/kWh; for SO2, it is 2.74 g/kWh, and for NO_x, it is 1.34 g/kWh [10,24,48]. In this study, all the pollutants were considered to be contributing to greenhouse gas emissions. In [87], the authors presented an overview of air pollution in the 21st century, including emissions from power generation. The authors [87] discussed the effects of air pollution on human health and the environment and examined various mitigation strategies, such as improving fuel efficiency and reducing emissions through the use of clean technologies. This study assumed that greenhouse emissions are mostly from thermal power plants.

The general equation for calculating greenhouse gas emissions [10] is

$$GH{G}_e=E\times G{H}_{ef}\times (1-G{H}_{ref})$$

(13)

where $GH{G}_e$ is the GHG emissions, $E$ is the total substation power capacity (kWh), $G{H}_{ef}$ is the greenhouse emission factor (g/kWh), and $G{H}_{ref}$ is the overall greenhouse reduction efficiency.

3.5.3. Cost of Energy Loss ($CEL$)

$CEL$ includes the losses in electricity that occur during its generation, transmission, and distribution. The cost of energy losses reflects the full economic, social, and environmental impacts of energy waste [88]. This economic factor relates to the total cost savings after the implementation of DCER. It depends on the active power loss and the annual cost factor constant. $CEL$ (per annum) [17] is calculated with the help of Equation (14).

$$CEL={\displaystyle \sum _{N=i}^N{P}_T}\times (r_p+r_e\times L_{sf}\times 8760)$$

(14)

where $P_T$ is the total active power loss, $r_p$ is the annual demand cost of power loss (57.6923 USD/kW), $r_e$ is the annual cost of energy loss (0.00961538 USD/kWh), and $L_{sf}$ is the loss factor.

The loss factor can be calculated as

$$L_{sf}=s\times l_f+(1-s)\times l_f^2$$

(15)

In this study, $s$ = 0.2 and the practical load factor $l_f$ was set to 0.47 [17].

3.5.4. Distribution Clean Energy Resource Cost

The cost parameters have an impact on the total cost of operating the distribution system. The active and reactive power ratings of a distributed generation resource determine the cost of DCER installation [17,19]. Designing DCER to integrate with any existing system is an important aspect. In this study, Equations (16) and (17) were used to calculate the total cost of the DCER based on the active and reactive power cost functions.

$$C(P_{dcer})=c_1\times P_{dcer}^2+c_2\times P_{dcer}+c_3$$

(16)

where $c_1,c_2,c_3$ are the cost coefficients which were set to 0, 20, and 0.25 [17].

$$C(Q_{dcer})=[\cos t(S_{g\max })-\cos t(\sqrt{(S_{g\max }^2-Q_g^2)}]\times k_q$$

(17)

• where $P_{dcer}$ and $Q_{dcer}$ are the active and reactive powers of DCER,
• where $S_{g\max }={\displaystyle \frac{P_{g\max }}{\cos \varphi }}$, and $P_{g\max }=1.1\times P_{dcer}$.

In this study, $\cos \varphi$ is the power factor and $k_q$ varied between 0.05 and 0.1 [19].

3.5.5. Payback Year for DCER

Distribution network investors often use payback year as a way to calculate investment returns. It helps in determining the duration that it will take to recover the initial investment costs. When making decisions, this indication is beneficial. In this study, cash flow per year was calculated in terms of CEL savings after the DCER integration. Payback year was calculated using Equation (18).

$$PB{Y}_{dcer}={\displaystyle \frac{C_{iic}^{dcer}}{CF{Y}_{cel\text{\_}S}}}$$

(18)

where $PB{Y}_{dcer}$ is the payback year for the DCER, $C_{iic}^{dcer}$ is the initial installation cost of the DCER, and $CF{Y}_{cel\text{\_}S}$ is the cash flow saved per year.

3.5.6. Solar Model

In a solar PV model, the active power is based on solar irradiance and temperature; the reactive power in a PV system can be managed using an inverter, which can be controlled to provide or absorb reactive power as needed. The amount of reactive power that can be provided is limited by the inverter’s capacity and the amount of active power being generated.

$$P_{pv}={\eta }_{pv}\times A_{pv}\times I\times (1-\beta (T_{cell}-T_{ref}))$$

(19)

where $P_{pv}$ is the power output, ${\eta }_{pv}$ is the efficiency, $A_{pv}$ is the area, $I$ is the irradiance, $\beta$ is the temperature coefficient, $T_{cell}$ is the cell temperature, and $T_{ref}$ is the reference temperature.

The reactive power $Q_{pv}$ provided by the inverter can be calculated as

$$Q_{pv}=\sqrt{S_i^2-P_{pv}^2}$$

(20)

where $S_i$ represents the inverter capacity, which is the apparent power (VA).

3.5.7. Wind Model

In a wind turbine model, the active power is based on the wind speed, air density, and swept area; the reactive power can be significant and depends on the type of generator and control strategies.

$$P_{wind}=0.5\times \rho \times A\times v^3\times {\eta }_{wind}$$

(21)

where $P_{wind}$ is the power output, $\rho$ is the air density, $A$ is the swept area, $v$ is the wind speed, and ${\eta }_{wind}$ is the efficiency.

4. Multi-Objective Performance Function

The multi-objective performance function (PFMO) was used to optimize the operation and control of the distribution system by considering multiple system performance metrics like active power, reactive power, and voltage deviation. The PF_MO was minimized using soft computing techniques to find the optimal solution for these performance metrics to increase the reliability and improve the overall performance of the network. This study applied the priority-based indices weight factor [21] in the multi-objective performance function. This $P{F}_{MO}$ function can be formulated as

$$P{F}_{MO}=w_1\times A{P}_{loss\text{\_}index}+w_2\times V{D}_{index}+w_3\times R{P}_{loss\text{\_}index}$$

(22)

where $w_1,w_2,w_3$ are the priority indexing coefficients, which were set to 0.45, 0.35, and 0.20 [21]. The values of parameters like the number of swarms, iterations, and trials used for each method were as follows: (i) the population size was 32 for the 33-bus system for each method, (ii) the population size was 117 swarms for the 118-bus distribution system for each method, and (iii) a maximum number of 500 iterations and 100 trials were considered for each method.

4.1. Active Power Loss Index ($A{P}_{loss\text{\_}index}$)

$$A{P}_{loss\text{\_}index}={\displaystyle \frac{P_{l_{with\text{\_}DCER}}^a+P_{l_{with\text{\_}DCER}}^b+P_{l_{with\text{\_}DCER}}^c}{P_{l_{base}}^a+P_{l_{base}}^b+P_{l_{base}}^c}}$$

(23)

where $P_{l_{with\text{\_}DCER}}^a,P_{l_{with\text{\_}DCER}}^b,P_{l_{with\text{\_}DCER}}^c$ are the active power losses with DCER integration and $P_{l_{base}}^a,P_{l_{base}}^b,P_{l_{base}}^c$ are the active power losses without DCER in the three phases, respectively.

4.2. Reactive Power Loss Index ($R{P}_{loss\text{\_}index}$)

$$R{P}_{loss\text{\_}index}={\displaystyle \frac{Q_{l_{with\text{\_}DCER}}^a+Q_{l_{with\text{\_}DCER}}^b+Q_{l_{with\text{\_}DCER}}^c}{Q_{l_{base}}^a+Q_{l_{base}}^b+Q_{l_{base}}^c}}$$

(24)

where $Q_{l_{with\text{\_}DCER}}^a,Q_{l_{with\text{\_}DCER}}^b,Q_{l_{with\text{\_}DCER}}^c$ are the reactive power losses with DCER and $Q_{l_{base}}^a,Q_{l_{base}}^b,Q_{l_{base}}^c$ are the reactive power losses without DCER in the three phases, respectively.

4.3. Voltage Deviation Index (VDI)

$$V{D}_{index}=\left(max({\displaystyle \frac{{\mathrm{V}}_{i\text{\_}a,ref}-{\mathrm{V}}_{i\text{\_}a}}{V_{i\text{\_}a,ref}}})+max({\displaystyle \frac{{\mathrm{V}}_{i\text{\_}b,ref}-{\mathrm{V}}_{i\text{\_}b}}{V_{i\text{\_}b,ref}}})+max({\displaystyle \frac{{\mathrm{V}}_{i\text{\_}c,ref}-{\mathrm{V}}_{i\text{\_}c}}{V_{i\text{\_}c,ref}}})\right)/3$$

(25)

where ${\mathrm{V}}_{i\text{\_}a},{\mathrm{V}}_{i\text{\_}b},{\mathrm{V}}_{i\text{\_}c}$ are the node voltages; i varies between 1 and 33; and ${\mathrm{V}}_{i\text{\_}a,ref},{\mathrm{V}}_{i\text{\_}b,ref},{\mathrm{V}}_{i\text{\_}a,ref}$ are the reference voltages of the respective ith bus in the three-phases.

5. Soft Computing Techniques

Soft computing optimization techniques were applied to the power distribution network to minimize or maximize the objective goal. Some of these optimization techniques are based on different intelligent behaviours of social organisms like birds, animals, etc. In this work, swarm intelligence and evolutionary computing-based soft computing optimization techniques were used to evaluate the performance of balanced and unbalanced distribution networks. Figure 3 shows a block diagram of the soft computing technique, which used the PSO, TLBO, JAYA, SCO, RAO, and HBO techniques for the comparative analysis.

5.1. Particle Swarm Optimization (PSO)

PSO was formulated by Eberhart in 1995 and is based on the social interaction of natural heuristic biological movements and the intelligent behaviours of swarms. In PSO [32], the developer consists of a group containing the swarm and target. Each swarm particle moves towards the target based on its own experience. Each swarm tracks the best search location, which is called the personal best. Then, all the swarms are compared in terms of the best fitness and location with respect to the assigned target, which is known as global best. To improve fitness, a weighted acceleration factor is introduced to each swarm. For a group of swarms, the next position and velocity of each swarm is updated using Equations (23) and (24).

$$x^i(n+1)=x^i(n)+v^i(n+1)$$

(26)

Then, the velocities are updated using Equation (22).

$$v_{i=n}^{n+1}=w\times v_{i=n}^n+(c_1\times r_1\times (x_{pbest}-x_{i=n}^n)+c_2\times r_2\times (x_{gbest}-x_{i=n}^n)$$

(27)

where w is the PSO weight factor; $r_1,r_2$ are random variables between 0 and 1; $c_1,c_2$ are PSO acceleration coefficients; and $x_{pbest},x_{gbest}$ are the personal and global best of each particle.

5.2. Teaching–Learning-Based Optimization

The TLBO algorithm [33,34,35] was formulated by Rao and is based on classroom teaching–learning processes. In TLBO, a group of students is considered a population. In the first teacher phase, the teacher delivers knowledge to the students, after which, each student develops their own knowledge-based abilities. In the learner phase, the students find the best student among them and start to learn by making partners. Therefore, in the learner phase, students check their partner’s fitness, and then decide whether to learn from their partner or give knowledge to their partner.

5.2.1. Phase I: Teaching Phase

If we consider $f$ to be the minimum fitness of a student, which is assumed by the teacher, and $f_{mean}$ is the mean value of all students, then

$$f_{new}=f+r(f_{teacher}-TF(f_{mean}))$$

(28)

where $f_{new}$ is the best fitness in the teacher phase and is the teaching factor, and r represents a random variable.

5.2.2. Phase II: Learner Phase

Selecting any partner randomly, whose fitness is denoted as $f_{partner}$, these partners interact and exchange knowledge, so the new fitness is

$$f_{new}=f\pm r(f-(f_{partner}))$$

(29)

5.3. JAYA Optimization Algorithm

The JAYA algorithm [36] is based on the identification of the best and worst solutions in constrained and unconstrained problems. It does not involve any algorithm-specific parameter. It is free from control parameters and simulates a single phase. The JAYA algorithm initially identifies the best and worst solutions and then modifies the best and worst solutions using Equation (27).

$$p_{i=n}^{n+1}=p_{i=n}^n+(r_1\times (p_{best}-\left|p_{i=n}^n\right|)-r_2\times (p_{worst}-\left|p_{i=n}^n\right|))$$

(30)

where $r_1$ and $r_2$ are random variables, which range between 0 and 1.

5.4. Sine Cosine Optimization (SCO) Algorithm

SCO is based on the stochastic population technique [37]. Exploration and exploitation are two phases in SCO. In the first phase, a random solution search is conducted with higher speed and in the second phase, the speed of the searching process is comparatively slower. Sine cosine optimization simulates the optimal desire value using Equation (28).

$$\begin{array}{l}{x}_{i,t+1}=\left\{\begin{array}{cc}{x}_{i,t}+n_1\times \sin (n_2)\times |n_3{P}_{i,t}-x_{i,t}|& n_4<0.5\\ x_{i,t}+n_1\times \cos (n_2)\times |n_3{P}_{i,t}-x_{i,t}|& n_4\ge 0.5\end{array}\right.\\ i=1,2,\dots ,n\end{array}$$

(31)

where $n_1=a-{\displaystyle \frac{a\times itr}{it{r}_m}}$; $itr$ and $it{r}_m$ are the current and maximum iterations; $a$ denotes the sine cosine constant ($a$ = 2); $n_1$ denotes the next position; $n_2$ denotes the movement towards the final outward destination; $n_3$ is the random weight for emphasis ($n_3$ > 1) or deemphasis ($n_3$ < 1); and $n_4$ is the switch between the sine and cosine components.

5.5. RAO Optimization Algorithm

RAO optimization techniques [38,39] are based on the best and worst optimal solutions, and they do not require any algorithm design with constant parameters. In RAO, the population size is initialized by a random variable with upper and lower boundary limits, and then the best and worst optimal solutions are identified. In this technique, the optimal solution is calculated with the help of Equation (29).

$$p_{i=n}^{n+1}=p_{i=n}^n+(r_1\times (p_{i,best}-p_{i,worst}))$$

(32)

where $r_1,r_2$ are random variables that range between 0 and 1. Moreover, $p_{i,best},p_{i,worst}$ are the best and worst solutions.

5.6. HBO Algorithm

The Honey Badger Optimization Algorithm [65,66] is a nature-inspired algorithm; it simulates the foraging behaviour of honey badgers, which are noted for their intelligence, tenacity, and adaptability in locating food. The method uses these properties to effectively explore and exploit the search space. Honey badgers locate food sources by smelling, digging, or following honeyguide birds. In the digging and honey mode, the animal uses its sense of smell to locate prey and then walks about to dig and catch it. Then, after the honey badgers practice, a honeyguide bird helps them to locate beehives.

5.6.1. Phase I: Digging Phase

$$H_{new}=H_{prey}+F\times \beta \times I\times H_{prey}+F\times r_1\times {\alpha }_{snf}\times S_i\times |\cos (2\pi r_2)\times [1-\cos (2\pi r_3)]$$

(33)

5.6.2. Phase II: Honey Phase

$$H_{new}=H_{prey}+F\times r_4\times {\alpha }_{snf}\times S_i$$

(34)

where $H_{new}$ is the position of new prey and $H_{prey}$ is the position of previous prey, $\beta$ is 6 [66], $S_i$ is the distance, $F$ is equal to 1 or −1, ${\alpha }_{snf}$ is the search factor, and $r_1,r_2,r_3,r_4$ are random variables between 0 and 1.

The flowchart in Figure 4 illustrates the step-by-step implementation of the proposed methodology using soft computing optimization techniques for both balanced and unbalanced loads. Backward and forward sweep techniques were used to compute the distribution system load flow.

6. Results and Discussion

Three-phase balanced and unbalanced load models using modified IEEE 33-bus distribution system data were used in the case I study. The standard IEEE 118-bus and modified IEEE 118-bus distribution systems were used in the case II study. Case II was used to validate and compare the performance results with that of the IEEE 33-bus test system. The performance metrics were computed using the forward and backward load flows. Six soft computing optimization strategies were implemented for DCER integration. HBO is the latest optimization, and PSO was the primary optimization method for this study; every optimization technique has been implemented with the same population size and number of iterations.

6.1. Case I: IEEE 33-Bus System

In this study, the minimum fitness factors were used to decide the optimal location and size of the DCER, which are shown in

Table 3. Optimal DCER size and location based on fitness factor for case I.

Optimization Technique	Modified IEEE 33 Balanced Case
	Multi-Objective Fitness Factor	Size and Location of DCER1			Size and Location of DCER2
		P (kW)	Q (kVAr)	Location	P (kW)	Q (kVAr)	Location
PSO	0.12553	1075.7813	507.9305	13	961.0633	1150.0000	30
TLBO	0.12201	846.0809	315.1442	13	1137.5810	1004.1470	30
JAYA	0.11984	849.1086	390.8586	13	1112.3944	1080.6349	30
SCO	0.11393	822.2383	351.1445	13	1102.8561	1066.9439	30
RAO	0.10857	834.9450	398.0917	13	1116.0595	1048.5717	30
HBO	0.10851	865.9414	421.9040	13	1103.9513	1039.1702	30
Optimization Technique	Modified IEEE 33 Unbalanced Case
	Multi-Objective Fitness Factor	P (kW)	Q (kVAr)	Location	P (kW)	Q (kVAr)	Location
PSO	0.2088	986.4774	897.1920	12	1215.0997	805.6353	30
TLBO	0.2064	723.6780	702.6870	11	1139.0659	780.3199	30
JAYA	0.1903	883.6662	413.1314	13	1152.4844	1148.5203	30
SCO	0.1885	958.1873	340.0362	12	1089.8410	1149.5499	30
RAO	0.1865	853.2926	397.2511	13	1151.1015	1116.8672	30
HBO	0.1826	883.6394	372.8694	13	1122.6246	1120.131	30

, using six different soft computing optimization techniques. Comparative optimizations were performed for the balanced and unbalanced load cases. HBO provided the best fitness value for PF_MO in both scenarios after comparing the optimizations. In the balanced case, all the above optimizations were for the location of the DCER, which were integrated on buses 13 and 30, but in the unbalanced case, the optimal DCER location of one DCER varied between buses 11, 12, and 13 using different optimizations, and the second DCER, in all optimizations, was located on bus 30. The installation cost of the DCER with different combinations is given in

Table 4. Installation cost of DCER for case I.

Optimization Technique	Modified IEEE 33 Balanced Case			Modified IEEE 33 Unbalanced Case
	P Cost (USD)	Q Cost (USD)	Total Cost (USD)	P Cost (USD)	Q Cost (USD)	Total Cost (USD)
PSO	33,158.85	1317.297	34,476.15	34,056.65	1353	35,409.65
TLBO	26,385.85	1048.225	27,434.08	29,660.23	1178	30,838.23
JAYA	29,430.01	1169.161	30,599.17	31,233.25	1241	32,474.25
SCO	28,361.85	1126.726	29,488.58	29,791.95	1184	30,975.95
RAO	28,933.47	1149.435	30,082.90	30,282.45	1203	31,485.45
HBO	29,221.73	1160.881	30,382.61	29,860.25	1186	31,046.25

. In this study, the location, size, and cost were the conclusive factors for deciding the final size and location of the DCER. After a comparative analysis of the balanced case and unbalanced case, HBO gave the best optimal value for the integration of DCER.

The first DCER could be installed at bus 13 with active and reactive power ratings of 865.94 kW and 421.90 kVAr for the DCER. The second DCER could be installed at bus 30 with active and reactive powers of 1103.95 kW and 1039.17 kVAr, respectively. Table 4 shows the installation cost of the DCER with different configurations. Here, TLBO gave the smallest installation cost but an integration of all the factors (cost, size, and location) occurs simultaneously, so in the comparative analysis, the HBO size, location, and cost in the unbalanced case were considered for the installation, so the final installation cost was USD 31,046.25.

The dependency on the central grid was reduced by 55–63% in both the balanced and unbalanced cases, which is presented in

Table 5. Power generation requirements from the central grid system for case I.

Optimization Technique	Modified IEEE 33 Balanced Case			Modified IEEE 33 Unbalanced Case
	Grid kVA without DCER	Grid kVA after DCER Installation	Grid Power Savings	Grid kVA without DCER	Grid kVA after DCER Installation	Grid Power Savings
PSO	4599.1	1832.4	60%	4645.6	1696.8	63%
TLBO		2022.6	56%		2092.8	55%
JAYA		1971.4	57%		1896.0	59%
SCO		2028.6	56%		1916.5	59%
RAO		1991.5	57%		1944.4	58%
HBO		1968.3	57%		1951.8	58%

, but the dependency after the final installation was reduced by 57% in the balanced case and 58% in the unbalanced case. These data show the impact of the utilization of renewable clean energy sources on saving grid energy, which can be utilized for future load extensions in the distribution network.

The annual cost of energy loss in the balanced case was USD 16,314.5 but after optimal DCER integration, it decreased to USD 2295.9 using the RAO technique, and all the other CEL savings varied between 84 and 86% in the balanced case. In the unbalanced case, the cost was USD 194,426.4 but after optimal DCER integration, it decreased to USD 4313.6 using RAO, but HBO also give a similar value (77.74%) in the unbalanced case, as shown in

Table 6. Annual costs of energy loss for case I.

Optimization Technique	Modified IEEE 33 Balanced Case					Modified IEEE 33 Unbalanced Case
	Total DCER Installation Cost (USD)	CEL (Cost of Energy Loss) (USD)	CEL with DCER (USD)	CEL Cost Savings per Annum in USD (%)	Payback Period for DCER	Total DCER Installation Cost (USD)	CEL (Cost of Energy Loss) (USD)	CEL with DCER (USD)	CEL Cost Savings per Annum in USD (%)	Payback Period for DCER
PSO	34,476.15	16,314.5	2558.3	13,756.2 (84.32%)	2.5	35,409.6	19,426.4	4856.6	14,569.9 (75%)	2.4
TLBO	27,434.08		2337.9	13,976.6 (85.67%)	2.0	30,838.5		4796.3	14,630.1 (75.31%)	2.1
JAYA	30,599.17		2295.7	14,018.9 (85.92%)	2.2	32,474.0		4319.2	15,107.2 (77.76%)	2.1
SCO	29,488.58		2378.7	13,935.8 (85.42%)	2.1	30,975.5		4364.5	15,061.9 (77.53%)	2.1
RAO	30,082.90		2295.9	14,018.7 (85.92%)	2.1	31,485.5		4313.6	15,112.9 (77.73%)	2.1
HBO	30,382.61		2299.8	14,014.6 (85.90%)	2.1	3146.25		4321.3	15,105.1 (77.74%)	2.1

. When the cost savings were compared with the total installation cost, it was found that within two and half years, the cost of the DCER will be compensated for through these savings. The annual CEL savings directly benefited the consumers and distribution companies.

Table 7. Environmental impact reduction factor for case I.

Power Generation at Slack Bus					Greenhouse Gas Emissions in g/kWh	Yearly Greenhouse Gas Emissions in Tonnes without DCER	Yearly Greenhouse Gas in Tonnes after DCER Installation	Emission Savings after Renewable DCER Installation
Optimization Technique	Modified IEEE 33 Balanced Case
	PG (kW)	QG (kVAr)	PG (kW)	QG (kVAr)
	Without DCER		With DCER
PSO	3905.44	2428.90	1708.00	663.66	632.4683	21,313.84	9463.06	56%
TLBO			1758.02	1000.19			9740.20	54%
JAYA			1779.78	847.74			9860.73	54%
SCO			1817.09	901.82			10,067.44	53%
RAO			1790.23	872.51			9918.64	53%
HBO			1771.41	858.13			9812.31	54%
Optimization Technique	Modified IEEE 33 Unbalanced Case				GHG Emissions in g/kWh	Yearly GHG in Tonnes without DCER	Yearly GHG Emissions in Tonnes after DCER Installation	Savings after Renewable DCER Installation
	PG	QG	PG	QG
	Without DCER		With DCER
PSO	3943.41	2455.80	1571.77	639.13	632.4683	21,521.07	8708.29	60%
TLBO			1908.98	857.69			10,576.53	51%
JAYA			1730.12	775.57			9585.59	55%
SCO			1718.75	847.89			9522.58	56%
RAO			1761.69	822.94			9760.51	55%
HBO			1759.89	844.04			9750.53	55%

shows the greenhouse gas (GHG) emissions calculated by the different soft computing optimization techniques. GHG emissions were reduced by 53–56% in the balanced case and 52–60% in the unbalanced case. The calculation assumed that the generator bus delivered power from the thermal power plant, and all pollutant emissions were spread out in the atmosphere. In this study, the pollutant emission factors for CO₂, SO₂, and NO_x were 632.0 g/kWh, 2.74 g/kWh, and 1.34 g/kWh, respectively. In this study, the total GHG emission using HBO optimization was the smallest. Thus, the GHG emission generated without DCER was 2152.07 tonnes per year and after DCER installation at buses numbers 13 and 30, it was reduced by 9750.53 tonnes per year in the unbalanced case; in the balanced case, it was 21,313.84 tonnes per year before the DCER installation, but once DCER were integrated, it was reduced by 9812.31 tonnes per year. The DCER integration not only improved the voltage profile but it also reduced greenhouse gas emissions, which directly benefits human and their ecological environment.

Figure 5 shows the comparative voltage profiles for each phase with PSO, TLBO, JAYA, SCO, and RAO in both the balanced and unbalanced load cases with and without DCER integration. Figure 6 shows the voltage profiles of the IEEE 33-bus system in the balanced case with the best optimization (HBO) and Figure 7 shows the three-phase voltage profiles in the unbalanced case with and without DCER integration.

Table 8. Minimum voltage without and with DCER for case I.

Optimization Technique	Modified IEEE 33 Balanced Case		Modified IEEE 33 Unbalanced Case
	Phase A/B/C Voltage (pu)	Bus No. (All Phases)	Phase A Voltage (pu)	Phase B Voltage (pu)	Phase C Voltage (pu)	Bus No. (A, B, C Phases)
Base Case without DCER	0.913	18	0.927	0.931	0.871	18, 18, 33
PSO with DCER	0.981	25	0.984	0.982	0.961	18, 18, 33
TLBO with DCER	0.98	25	0.982	0.98	0.952	25, 25, 18
JAYA with DCER	0.98	25	0.983	0.981	0.962	25, 25, 33
SCO with DCER	0.98	25	0.983	0.981	0.957	25, 25, 18
RAO with DCER	0.98	25	0.983	0.981	0.96	25, 25, 33
HBO with DCER	0.98	25	0.982	0.981	0.961	25, 25, 33

and

Table 9. Maximum voltage without and with DCER for case I.

Optimization Technique	Modified IEEE 33 Balanced Case		Modified IEEE 33 Unbalanced Case
	Phase A/B/C Voltage (pu)	Bus No. (All Phases)	Phase A Voltage (pu)	Phase B Voltage (pu)	Phase C Voltage (pu)	Bus No. (A, B, C Phases)
Base Case without DCER	1	1	1	1	1	1, 1, 1
PSO with DCER	1.013	13	1.022	1.02	0.961	30, 30, 33
TLBO with DCER	1	1	1.015	1.012	1	30, 30, 1
JAYA with DCER	1.001	13	1.022	1.02	1	30, 30, 1
SCO with DCER	1	1	1.021	1.019	1	30, 30, 1
RAO with DCER	1	1	1.021	1.019	1	30, 30, 1
HBO with DCER	1.001	13	1.021	1.019	1	30, 30, 1

show the minimum and maximum voltages at the different nodes. In the balanced case, the minimum voltage in bus 18 was 0.913 pu, but in the unbalanced case, the minimum voltages in phase A, phase B, and phase C were 0.927, 0.931, and 0.871 pu, which are less than the specified standard and create unbalancing. This unbalancing can be minimized by integrating DCER. Figure 5 and Figure 6 show the voltage profiles with different optimization techniques; with TLBO, the voltage was marginally higher in each bus compared with the others. Finally, HBO and RAO gave a better voltage profile in each node after the DCER integration.

The phase angle deviation with PSO, TLBO, JAYA, SCO, and RAO in the balance and unbalanced load cases are presented in Figure 8. From the figure, it can be observed that in the unbalanced case, the phase angle deviation varied more than in the balanced case. We can conclude that the nonlinearity behaviour increased due to an unbalanced network. As shown in Figure 9, in phase C, the maximum 2.4-degree phase angle distortion occurred in the unbalanced case. This distortion creates a heating effect in the polyphase-connected load, and consumers may face poor power quality problems and energy consumption. Unbalanced phase angle distortion can reduce electrical equipment’s lifespan and electrical systems’ overall reliability. Here, HBO produced the best results in the comparative analysis of the angle deviation for each phase, which is shown in Figure 9 and Figure 10, under both balanced and unbalanced load conditions.

Active and reactive power losses are unwanted losses in distribution systems. In this study, for each phase, the active loss was 67.56 kW and the total loss was 202.68 kW in the balanced load case, while in the unbalanced case, the active power loss was 45.54 kW, 45.88 kW, and 149.92 kW in each phase, respectively, and the total loss increased by 241.34 kW. Simultaneously, the reactive power increased in the unbalanced case; its value was 45.05 kVAr for each phase in the balanced case, and the total was 135.14 kVAr. In the unbalanced case, the reactive power losses in phase A, phase B, and phase C were 30.68, 29.87, and 101.84 kVAr, and the total loss increased up to 162.39 kVAr. After the DCER integration using the different optimization techniques, the active power loss was reduced by 84–86% in the balanced case and 75–78% in the unbalanced case, as shown in

Table 10. Active power loss using different optimizations for case I.

Optimization Technique	Active Power Losses without and with DCER (kW)
	Without DCER in Balanced Load Case (kW)				With DCER Integration in Balanced Load Case (kW)				Percentage Savings
	Phase A	Phase B	Phase B	Total	Phase A	Phase B	Phase C	Total
PSO	67.56	67.56	67.56	202.68	10.59	10.59	10.59	31.78	84%
TLBO					9.68	9.68	9.68	29.04	86%
JAYA					9.51	9.51	9.51	28.52	86%
SCO					9.85	9.85	9.85	29.55	85%
RAO					9.51	9.51	9.51	28.52	86%
HBO					9.47	9.47	9.47	28.40	86%
Optimization Technique	Without DCER in Unbalanced Load Case (kW)				With DCER Integration in Unbalanced Load Case (kW)				Percentage Savings
	Phase A	Phase B	Phase C	Total	Phase A	Phase B	Phase C	Total
PSO	45.54	45.88	149.92	241.34	14.81	16.59	28.93	60.33	75%
TLBO					12.29	13.18	34.11	59.58	75%
JAYA					14.53	15.60	23.52	53.66	78%
SCO					13.63	14.74	25.84	54.22	78%
RAO					13.94	14.93	24.72	53.59	78%
HBO					13.84	14.87	24.81	53.52	78%

. The reactive power loss was also reduced by 83–85% in the balanced case and 74–76% in the unbalanced case (

Table 11. Reactive power loss using different optimization techniques for case I.

Optimization Technique	Reactive Power Losses—Base Case and with DCER (kVAr)
	Without DCER in Balanced Load Case (kVAr)				With DCER Integration in Balanced Load Case (kVAr)				Percentage Savings
	Phase A	Phase B	Phase C	Total	Phase A	Phase B	Phase C	Total
PSO	45.05	45.05	45.05	135.14	7.53	7.53	7.53	22.58	83%
TLBO					6.89	6.89	6.89	20.68	85%
JAYA					6.79	6.79	6.79	20.37	85%
SCO					7.04	7.04	7.04	21.11	84%
RAO					6.78	6.78	6.78	20.33	85%
HBO					6.73	6.73	6.73	20.18	85%
Optimization Technique	Without DCER in Unbalanced Load Case (kVAr)				With DCER Integration in Unbalanced Load Case (kVAr)				Percentage Savings
	Phase A	Phase B	Phase C	Total	Phase A	Phase B	Phase C	Total
PSO	30.68	29.87	101.84	162.39	10.84	11.64	20.5	42.97	74%
TLBO					10.42	11.58	20.16	42.16	74%
JAYA					9.32	9.91	19.21	38.44	76%
SCO					9.87	10.01	19.13	39.01	76%
RAO					9.32	9.87	19.14	38.33	76%
HBO					9.29	9.87	19.05	38.21	76%

).

Table 12. Comparative analysis of active and reactive power losses with those of existing studies for case I.

IEEE 33 Balanced Case					Modified IEEE 33 Unbalanced Case
Case	Ploss (kW)	Ploss Reduction (%)	Qloss (kVAr)	Qloss Reduction (%)	Ploss (kW)	Ploss Reduction (%)	Qloss (kVAr)	Qloss Reduction (%)
Base Case	211.7	-	143.1	-	-	-	-	-
With DG/DCER	96.76 [89]	52.26%	NA	NA	NA	NA	NA	NA
	67.95 [90]	67.79%	54.79	61.69%	NA	NA	NA	NA
	139.53 [91]	33.87%	NA	NA	NA	NA	NA	NA
Base Case [92]	213	-	143	-	-	-	-	-
Case I [92]	112.3 [92]	47.27%	79.1	44.68%	NA	NA	NA	NA
Case II [92]	134 [92]	37.08%	90	37.07%	NA	NA	NA	NA
Base Case [59]	202.68 [59]	-	135.16 [59]	-	-	-	-	-
Proposed Work
Base Case	202.68	-	135.14	-	241.34	-	162.39	-
PSO with DCER	31.78	84%	22.58	83%	60.33	75%	42.97	74%
TLBO with DCER	29.04	86%	20.68	85%	59.58	75%	42.16	74%
JAYA with DCER	28.52	86%	20.37	85%	53.66	78%	38.44	76%
SCO with DCER	29.55	85%	21.11	84%	54.22	78%	38.72	76%
RAO with DCER	28.52	86%	20.33	85%	53.59	78%	38.33	76%
HBO with DCER	28.40	86%	20.18	85%	53.52	78%	38.21	76%

shows the results of the validation with the results from different existing works. We can conclude from the comparative study that HBO found the best optimal solution. In the balanced case, HBO estimated an active loss of 9.47 kW for each phase, and the total loss was 28.40 kW, while the reactive loss for each phase was 6.73 kVAr, and the total loss was 20.18 kVAr. In the unbalanced case, HBO calculated the active power losses in phase A, phase B, and phase C to be 13.84, 14.87, and 24.81 kW, respectively, and the total loss increased to 53.52 kW. The reactive power losses in phase A, phase B, and phase C were 9.29, 9.87, and 19.05 kVAr, respectively, and the total loss was 38.21 kVAr. Figure 11, Figure 12, Figure 13 and Figure 14 show the active and reactive power loss profiles in each branch using HBO in both cases.

6.2. Case II: IEEE 118-Bus Distribution System

In order to validate and demonstrate the performance in a higher bus system, a second case test, the IEEE 118-bus distribution system, was used in this study. The multi-objective fitness factor was used to determine the optimal location and size of the DCER, which is shown in

Table 13. Optimal DCER size and location based on fitness factor for case II.

Optimization Technique	IEEE 118 Balanced Case
	Multi-Objective Fitness Factor	Size and Location of DCER 1			Size and Location of DCER 2
		P (kW)	Q (kVAr)	Location	P (kW)	Q (kVAr)	Location
HBO	0.3680	2497.40	3068.91	109	2869.75	2782.34	71
RAO	0.3734	3602.67	3339.05	107	2666.03	2780.68	72
TLBO	0.3926	3381.08	2743.44	109	2738.74	1850.33	72
SCO	0.3958	1859.20	1948.81	110	2917.40	1929.27	72
JAYA	0.3994	3461.27	2651.53	109	2798.93	1846.97	72
PSO	0.4163	2282.19	2653.46	112	2015.51	1692.03	76
Optimization Technique	Modified IEEE 118 Unbalanced Case
	Multi-Objective Fitness Factor	P (kW)	Q (kVAr)	Location	P (kW)	Q (kVAr)	Location
HBO	0.4421	6021.13	3724.61	107	3124.45	3212.12	72
TLBO	0.4427	3514.14	2630.10	109	3301.42	1152.42	72
JAYA	0.4441	3314.66	2443.87	108	2719.22	2040.63	72
RAO	0.4496	3249.56	2125.19	108	2312.22	1796.60	72
PSO	0.4530	1823.25	1550.30	118	3184.22	1800.42	72
SCO	0.4588	2063.26	2232.36	118	3259.64	1760.95	72

, using the different soft optimization techniques. HBO showed the best fitness factor among the optimization techniques in both cases. The optimal size and location are shown in Table 13, and the installation cost of the DCER with the different combinations is given in

Table 14. Installation cost of DCER for case II.

Optimization Technique	IEEE 118 Balanced Case			Modified IEEE 118 Unbalanced Case
	P Cost (USD)	Q Cost (USD)	Total Cost (USD)	P Cost (USD)	Q Cost (USD)	Total Cost (USD)
PSO	86,910.05	3452.68	90,362.73	67,014.65	2662.29	69,676.94
TLBO	91,875.65	3649.95	95,525.60	75,650.65	3005.38	78,656.03
JAYA	74,643.45	2965.36	77,608.81	89,690.25	3563.13	93,253.38
SCO	60,315.25	2396.14	62,711.39	79,866.45	3172.86	83,039.31
RAO	87,752.65	3486.15	91,238.80	78,436.05	3116.03	81,552.08
HBO	117,025.25	4649.07	121,674.32	138,734.85	5511.53	144,246.38

.

Table 15. Power generation requirements from the central grid system for case II.

Optimization Technique	IEEE 118 Balanced Case			Modified IEEE 118 Unbalanced Case
	Grid kVA without DCER	Grid kVA after DCER Installation	Grid Power Savings	Grid kVA without DCER	Grid kVA after DCER Installation	Grid Power Savings
PSO	29,910.31	23,145.90	23%	30,321.94	21,022.24	31%
TLBO		21,428.85	28%		21,709.28	28%
JAYA		21,374.02	29%		21,876.55	28%
SCO		22,981.19	23%		19,818.67	35%
RAO		20,486.81	32%		22,607.49	25%
HBO		21,371.72	29%		18,057.45	40%

illustrates the power generation requirements from the central grid system and it presents the reduction in the reliance on the central grid in both the balanced and unbalanced cases. For the final installation, the reduction was 23–33% in the balanced case and 25–40% in the unbalanced case. Here, HBO produced in better results, with 29% savings in the balanced case and 40% in the unbalanced case.

As shown in

Table 16. Annual costs of energy loss for case II.

Optimization Technique	IEEE 118 Balanced Case					Modified IEEE 118 Unbalanced Case
	Total DCER Installation Cost (USD)	CEL (Cost of Energy Loss) (USD)	CEL with DCER (USD)	CEL Cost Savings per Annum in USD (%)	Payback Period for DCER	Total DCER Installation Cost (USD)	CEL (Cost of Energy Loss) (USD)	CEL with DCER (USD)	CEL Cost Savings per Annum in USD (%)	Payback Period for DCER
PSO	90,362.73	104,180.32	51,316.7 (50.7%)	52,863.54	1.97	69,676.94	132,275.25	68,305.92	63,969.3 (48.4%)	2.07
TLBO	95,525.60		44,418.5 (57.4%)	59,761.79	1.74	78,656.03		66,109.85	66,165.3 (50.0%)	2.00
JAYA	77,608.81		44,417.7 (57.4%)	59,762.56	1.74	93,253.38		65,449.08	66,826.1 (50.5%)	1.98
SCO	62,711.39		47,786.9 (54.1%)	56,393.37	1.85	83,039.31		72,430.65	59,844.6 (45.2%)	2.21
RAO	91,238.80		49,050.6 (52.9%)	55,129.66	1.89	81,552.08		66,431.78	65,843.4 (49.8%)	2.01
HBO	121,674.32		47,653.6 (54.3%)	56,526.69	1.84	144,246.38		76,447.14	55,828.1 (42.2%)	2.37

, the annual cost of the energy loss in the IEEE 118 system was USD 104,180.32, which was reduced by about 50.7–57.4% in the balanced case and by 42.2–50.5% in the unbalanced case. The payback period for the DCER was around two years in the balanced case and more than two years in the unbalanced case. However, in both cases, the cost of the loss decreased when utilizing integrated DCER, and these energy savings can be used to compensate for the DCER installation cost.

Table 17. Environmental impact reduction.

Power Generation in Slack Bus					Greenhouse Gas Emissions in g/kWh	Yearly Greenhouse Gas Emissions in Tonnes without DCER	Yearly Greenhouse Gas Emissions in Tonnes after DCER Installation	Emission Savings after Renewable DCER Installation
Optimization Technique	IEEE 118 Balanced Case
	PG (kW)	QG (kVAr)	PG (kW)	QG (kVAr)
	Without DCER		With DCER
PSO	23,980.84	17,875.85	19,057.37	13,135.81	632.47	132,863.955	105,586.1439	21%
TLBO			17,154.68	12,841.82			95,044.44881	28%
JAYA			17,014.43	12,936.68			94,267.39473	29%
SCO			18,533.78	13,588.01			102,685.2311	23%
RAO			17,065.27	11,335.16			94,549.04072	29%
HBO			17,946.67	11,604.63			99,432.42484	25%
Optimization Technique	Modified IEEE 118 Unbalanced Case				GHG Emissions in g/kWh	Yearly GHG Emissions in Tonnes without DCER	Yearly GHG Emissions in Tonnes after DCER Installation	Savings after Renewable DCER Installation
	PG	QG	PG	QG
	Without DCER		With DCER
PSO	24,320.14	18,109.41	16,734.46	12,723.69	632.47	134,743.8705	92,716.22876	31%
TLBO	24,320.14	18,109.41	16,720.77	13,845.89			92,640.40031	31%
JAYA	24,320.14	18,109.41	17,493.72	13,135.95			96,922.85679	28%
SCO	24,320.14	18,109.41	16,231.89	11,371.25			89,931.75494	33%
RAO	24,320.14	18,109.41	17,976.07	13,709.82			99,595.31055	26%
HBO	24,320.14	18,109.41	14,524.85	10,728.48			80,474.03852	40%

shows that the greenhouse gas (GHG) emissions were reduced by 21–29% in the balanced case and 26–40% in the unbalanced load case. HBO gave better results in the unbalanced case, while the RAO and JAYA techniques showed better results under balanced load conditions. Figure 15 and Figure 16 show the voltage profiles with PSO for each phase in the balanced and unbalanced cases. Figure 17 and Figure 18 show the voltage profiles with TLBO for each phase in the balanced and unbalanced cases. Figure 19 and Figure 20 show the voltage profiles with JAYA optimization for each phase in the balanced and unbalanced cases. Figure 21 and Figure 22 show the voltage profiles with SCO for each phase in the balanced and unbalanced cases. Figure 23 and Figure 24 show the voltage profiles with RAO for each phase in the balanced and unbalanced cases. Figure 25 and Figure 26 show the voltage profiles with PSO for each phase in the balanced and unbalanced cases with and without DCER integration.

Table 18. Minimum voltage without and with DCER for case II.

Optimization Technique	IEEE 118 Balanced Case		Modified IEEE 118 Unbalanced Case
	Phase A/B/C Voltage (pu)	Bus No. (All Phases)	Phase A Voltage (pu)	Phase B Voltage (pu)	Phase C Voltage (pu)	Bus No. (A, B, C Phases)
Base Case without DCER	0.872069	76	0.857238	0.798224	0.879212	111, 76, 76
PSO with DCER	0.907167	54	0.925452	0.851135	0.898199	111, 43, 54
TLBO with DCER	0.907167	54	0.931914	0.851135	0.898199	46, 43, 54
JAYA with DCER	0.907167	54	0.931914	0.851135	0.898199	46, 43, 54
SCO with DCER	0.907167	54	0.931914	0.851135	0.898199	46, 43, 54
RAO with DCER	0.907167	54	0.931914	0.851135	0.898199	46, 43, 54
HBO with DCER	0.907167	54	0.931914	0.851135	0.898199	43, 46, 54

and

Table 19. Maximum voltage without and with DCER for case II.

Optimization Technique	IEEE 118 Balanced Case		Modified IEEE 118 Unbalanced Case
	Phase A/B/C Voltage (pu)	Bus No. (All Phases)	Phase A Voltage (pu)	Phase B Voltage (pu)	Phase C Voltage (pu)	Bus No. (A, B, C Phases)
Base Case without DCER	1.00000	1	1.00000	1.00000	1.00000	1, 1, 1
PSO with DCER	1.00000	1	1.04978	1.02972	1.02868	72, 117, 117
TLBO with DCER	1.00061	109	1.04300	1.00000	1.04714	72, 01, 109
JAYA with DCER	1.00129	109	1.04172	1.00000	1.03829	72, 01, 108
SCO with DCER	1.00518	72	1.05106	1.03835	1.04143	72, 117, 118
RAO with DCER	1.01135	72	1.02779	1.00000	1.03489	72, 01, 108
HBO with DCER	1.01178	71	1.06793	1.01357	1.07065	72, 107, 107

illustrate the minimum and maximum voltages at different nodes in case II with the IEEE 118 system. In the balanced load case, the minimum voltage at bus number 76 was 0.872069 pu under the base condition, but with DCER integration, it improved to 0.907167 at bus number 54. However, in the unbalanced case, the minimum voltages in phases A, B, and C were 0.857238, 0.798224, and 0.879212 pu, respectively. After integration, the DCER minimum voltage improved in all phases. In Table 19, the maximum voltage improved in both cases. Figure 27 and Figure 28 show the angle deviation for each phase in the balanced and unbalanced cases with HBO, which shows the best multi-objective fitness function value.

Table 20. Active power loss using different optimization techniques for case II.

Optimization Technique	Active Power Losses without and with DCER (kW)
	Without DCER in Balanced Load Case (kW)				With DCER Integration in Balanced Load Case (kW)				Percentage Savings
	Phase A	Phase B	Phase C	Total	Phase A	Phase B	Phase C	Total
PSO	431.41	431.41	431.41	1294.24	212.50	212.50	212.50	637.51	51%
TLBO					183.94	183.94	183.94	551.82	57%
JAYA					183.94	183.94	183.94	551.81	57%
SCO					197.89	197.89	197.89	593.66	54%
RAO					203.12	203.12	203.12	609.36	53%
HBO					197.34	197.34	197.34	592.01	54%
Optimization Technique	Without DCER in Unbalanced Load Case (kW)				With DCER Integration in Unbalanced Load Case (kW)
	Phase A	Phase B	Phase C	Total	Phase A	Phase B	Phase C	Total	Percentage Savings
PSO	436.54	881.55	325.18	1643.27	212.08	399.28	237.21	848.57	48%
TLBO					198.87	375.60	246.82	821.29	50%
JAYA					203.58	381.66	227.84	813.08	51%
SCO					216.74	406.40	276.68	899.81	45%
RAO					199.69	403.67	221.93	825.29	50%
HBO					254.22	375.08	320.40	949.71	42%

and

Table 21. Reactive power loss using different optimization techniques for case II.

Optimization Technique	Reactive Power Losses—Base Case and with DCER (kVAr)
	Without DCER in Balanced Load Case (kVAr)				With DCER Integration in Balanced Load Case (kVAr)				Percentage Savings
	Phase A	Phase B	Phase C	Total	Phase A	Phase B	Phase C	Total
PSO	325.09	325.09	325.09	975.28	185.79	185.79	185.79	557.36	43%
TLBO					169.21	169.21	169.21	507.63	48%
JAYA					169.02	169.02	169.02	507.06	48%
SCO					180.08	180.08	180.08	540.23	45%
RAO					175.08	175.08	175.08	525.25	46%
HBO					175.68	175.68	175.68	527.03	46%
Optimization Technique	Without DCER in Unbalanced Load Case (kVAr)				With DCER Integration in Unbalanced Load Case (kVAr)				Percentage Savings
	Phase A	Phase B	Phase C	Total	Phase A	Phase B	Phase C	Total
PSO	292.93	656.95	263.62	1213.49	154.64	356.34	187.72	698.70	42%
TLBO					151.88	353.57	198.49	703.95	42%
JAYA					153.03	356.81	185.99	695.82	43%
SCO					134.25	336.62	199.29	670.16	45%
RAO					152.35	371.61	185.03	708.99	42%
HBO					167.50	340.87	227.78	736.14	39%

list the active and reactive power losses for case II. According to the tables, the active power loss in all phases was 431.41 kW, and the total loss in the balanced load case was 1294.24 kW. On the other hand, the active power losses in the unbalanced case were 436.54 kW, 881.55 kW, and 325.18 kW in all the phases, respectively, and the total loss increased by 1643.27 kW. The reactive power increased simultaneously in the unbalanced case; in the balanced case, it was 325.09 kVAr for each phase, for a total of 975.28 kVAr. The reactive power losses in phase A, phase B, and phase C in the unbalanced load case were 292.93, 656.95, and 263.62 kVAr, and the overall loss was 1213.49 kVAr. Figure 29 and Figure 30 show the active power loss profiles for each phase in both the balanced and unbalanced cases and Figure 31 and Figure 32 show the reactive power loss profiles for each phase in the balanced and unbalanced cases with HBO.

7. Conclusions

The proposed method was tested using the modified IEEE 33-bus system under balanced and unbalanced conditions, and the larger IEEE 118-bus system was used to validate the performance results. Soft computing optimization techniques were used to optimize the location and size of the DCER. The performance of PSO, TLBO, JAYA, SCA, RAO, and HBO for both three-phase balanced and unbalanced distribution systems have been compared. Among them, the HBO and RAO techniques had the best multi-objective fitness factors: 0.1081 and 0.1085 in the balanced case and 0.1851 and 0.1865 in the unbalanced IEEE 33-bus network. In the IEEE 118 system, the multi-objective fitness factors were 0.3680 and 0.4421, respectively, in the balance and unbalanced load cases. DCER integration reduced the active and reactive power losses by 86% in the balanced case and 78% in the unbalanced case, but in previous studies, the active and reactive power losses were reduced by up to 67.79% and 61.69%, respectively, for a balanced 33-bus network (Table 12). GHG emissions, which is a pollutant emission factor, were reduced by all optimization techniques following the DCER implementation in the proposed system in both scenarios. In the IEEE 33-bus study, the PSO technique gave the most effective reduction in percentage and reduced emissions by 21,313.84 tonnes per year to 9463.06 in the balanced case and by 21,521.07 tonnes to 8708.29 tonnes in the unbalanced case. But, when all the performance parameters were considered, HBO gave the best solution for the placement of the DCER. The GHG emissions were reduced by 53% in the balanced scenario and 55% in the unbalanced case. Under both conditions, the GHG emissions were reduced in the IEEE 118-bus system, but the percentage was lower than in the IEEE 33-bus system. This occurred because only two DCER were employed in this study to validate the results. In order to achieve the most significant results in IEEE 118, the number of DCER integrations should be increased. The annual cost of the energy loss was reduced by 86% in the balanced case and by 78% in the unbalanced network after the DCER integration in the test bus network. This reduction in the annual cost could offset the total DCER installation cost. This study concludes that integrating DCER into distribution systems improves the utilization of clean energy resources, decreases the stress on the central grid, and improves the voltage profile of the network. The comparative analysis showed the technical, economic, and environmental impacts of balanced and unbalanced networks. In the future, our method could be used to analyse power quality issues, reliability, and resiliency with advanced modelling of DCER planning, including different practical loads and modern electric vehicle loads in the network.