Optimal allocation of solar photovoltaic distributed generation in electrical distribution networks using Archimedes optimization algorithm

Abstract

This paper proposes to resolve optimal solar photovoltaic (SPV) system locations and sizes in electrical distribution networks using a novel Archimedes optimization algorithm (AOA) inspired by physical principles in order to minimize network dependence and greenhouse gas (GHG) emissions to the greatest extent possible. Loss sensitivity factors are used to predefine the search space for sites, and AOA is used to identify the optimal locations and sizes of SPV systems for reducing grid dependence and GHG emissions from conventional power plants. Experiments with composite agriculture loads on a practical Indian 22-bus agricultural feeder, a 28-bus rural feeder and an IEEE 85-bus feeder demonstrated the critical nature of optimally distributed SPV systems for minimizing grid reliance and reducing GHG emissions from conventional energy sources. Additionally, the voltage profile of the network has been enhanced, resulting in significant reductions in distribution losses. The results of AOA were compared to those of several other nature-inspired heuristic algorithms previously published in the literature, and it was observed that AOA outperformed them in terms of convergence and redundancy when solving complex, non-linear and multivariable optimization problems.

Optimal solar photovoltaic system locations and sizes in electrical distribution networks are derived using a novel Archimedes optimization algorithm in order to minimize network dependence and pollutant emissions to the greatest extent possible.

solar photovoltaic system, electrical distribution network, composite load modelling, Archimedes optimization algorithm, distributed generation

Introduction

Traditional grid-scale power plants are incapable of meeting the growing demand for energy as fossil-fuel supplies decrease. Additionally, rising global temperatures have become a major impetus for the adoption of various renewable energy (RE) sources across the world [1] such as solar photovoltaic systems (SPVs), wind turbines (WTs), small-scale hydroelectric power plants, microturbines (MTs), geothermal power plants (GTPs), biomass systems (BSs) and fuel cells (FCs), among others, even at small-scale consumer sites as distributed generation (DG) units in electrical distribution networks (EDNs) [2]. Additionally, expanding and separating various types of loads, such as rural and urban, transportation, electric vehicles, municipalities, agricultural and other farms, can improve the overall performance and dependability of the distribution network. Agricultural consumers/feeders are currently supplied by traditional power plants and are thus permitted to use subsidized power at night. This situation increases the load on the grid. To mitigate this condition, solarization at the feeder level, especially for agricultural feeders in India, is gaining attention [3]. Modern EDNs require that such independent feeders be modelled, examined and optimized in order to function properly. Not only can RE-based DGs benefit the environment, but they can also benefit EDNs technically by reducing distribution losses, improving voltage profiles, enhancing stability margins and reducing grid dependence [4]. However, these benefits are only possible when EDNs, particularly radial distribution systems (RDSs), are correctly integrated [5].

The subject of optimum allocation of RE-based DGs (OADG) in EDNs has received a lot of attention in the literature, with diverse techno-economic-environmental goals, such as minimizing losses, voltage deviation index reduction, branch current reduction, improving voltage stability, annual cost (including costs of installation, operations and maintenance), reduction of greenhouse gas (GHG) emissions and power-quality consideration issues [6]. In the OADG problem, determining optimal locations (discrete variables) and selecting appropriate sizes (continuous variables) require a large computing impact as the network size grows. Furthermore, the OADG issue necessitates the fulfilment of multiple operational and planning aspects (unequal constraints) as well as energy balancing (equal constants), making it a more complex non-linear optimization problem. In contrast to the analytical approaches (AAs) [7], heuristic approaches (HAs) have been widely used to solve the OADG problem. In particular, HAs are more popular than AAs because they are simple to understand, easy to implement, derivative-free and good for multi-type and multidimensional optimization problems [8].

In [9], a modified symbiotic organisms search (SOS) with a chaos-based crossover operator at its parasitism phase is proposed for solving the multi-objective OADG problem. Three types of RE-based DGs (PV, WTs and BSs) are optimally allocated in EDNs for minimizing annual energy losses considering different seasons, installation and operational cost of DGs, annual grid energy purchase cost and voltage deviation. In [10], the manta ray foraging optimization algorithm (MRFO) is adapted for integrating real power compensators (i.e. Type-1 DGs like FC, PV and MTs). In [11], an improved variant of the marine predators algorithm (IMPA) is developed with modifications in exploration and exploitation phases embedded with multi-verse optimizer for improving the overall performance of EDN in terms of reduced losses, improved voltage profile and enhanced voltage stability via optimally locating and sizing the active and reactive power-based DGs. In [12], a multi-objective sine–cosine algorithm (SCA) with self-adaptive levy mutation and exponential conversion parameter is introduced for solving the OADG problem focusing on techno-economic-environmental aspects in the operation of the EDN and control under varying load profiles. In [13], a new pathfinding algorithm (PFA) based on optimal integration of a single large PV system is presented for improving the performance and resilience of EDN operation. A spotted hyena optimizer is used for solving active and reactive power-injection types of DGs in EDNs by aiming techno-economic benefits [14]. In [15], the artificial bee colony (ABC) is used for optimizing multi-objectives including overall operating cost, energy losses and voltage profile while solving the OADG problem. In [16], simultaneous allocation of active and reactive power-type DGs and dynamic network reconfiguration is solved for loss minimization and to reduce emissions under daily load-varying loading conditions using the jellyfish search algorithm (JFSA). In [17], the student psychology-based optimization algorithm is introduced for the first time to handle OADG problems by optimizing the performance of EDNs considering different types of load models. In [18], the ant-colony-optimization-based OADG problem is presented for loss minimization and voltage-profile improvement. In [19], the mutualism phase of basic SOS is modified with cloud-based theories (CMSOS) and used to solve a DG planning study by aiming technical benefits in EDNs. In [20], grey wolf optimization (GWO), MRFO, satin bower bird optimization and the whale optimization algorithm (WOA) are proposed for identifying the locations and sizes of DGs using Monte Carlo simulation-based probabilistic load flow for handling uncertainty of RE sources. Similarly, hybrid enhanced grey wolf optimizer and particle swarm optimization (EGWO-PSO) [21], harris hawks optimizer (HHO) [22], ant lion optimization (ALO) with particle swarm optimization (PSO) [23], artificial ecosystem-based optimization [24], PSO and butterfly optimization algorithm (BOA) [25], shuffled frog leap algorithm with binary PSO (BPSO-SLFA) [26], GWO [27], tunicate swarm intelligent algorithm (TSA) [28], enhanced genetic algorithm (EGA) [29] and future search algorithm (FSA) [30], firefly algorithm (FA) [31], enhanced sunflower optimization (ESFO) [32] and modified tabu search and harper sphere search (MTS-HSSA) [33] are some of such recent efficient metaheuristic algorithms proposed for the OADG problem.

According to the no-free-lunch theorem [34], no one algorithm can be used to solve all types of optimization issues; hence, academics are still motivated to develop new heuristic algorithms to solve optimization problems. Recent metaheuristic applications in the energy field include the honey badger algorithm (HBA) [35], modified artificial electric field algorithm (MAEF) [36], evaporation rate water cycle algorithm (ERWCA) [37], MRFO (MRF) [38], improved manta ray foraging optimizer [39], modified marine predator algorithm (MMPA) [40], equilibrium optimizer (EO) [41], turbulent flow of water-based optimization (TFWO) [42] and the gradient-based optimizer (GBO) [43].

In 2021, the Archimedes optimization algorithm (AOA) was a new successful metaheuristic algorithm by Hashim et al. [44] that balanced exploration and exploitation and demonstrated promising results for difficult optimization problems. The search efficiency of AOA was compared against well-known algorithms such as GA, PSO, L-SHADE and LSHADE-EpSin, as well as newer additions such as WOA, SCA, HHO and EO. As a result, the authors attempted to claim it as the first application to solve optimal renewable DG allocation in EDNs. AOA has recently attracted interest for a variety of optimization problems, including model parameter estimation for a proton exchange membrane (PEM) fuel cell [45], maximum power point tracking (MPPT) for a WT generator [46], simultaneous network reconfiguration (NR) and DG allocation [47], soft-open-points allocation along with NR and DG allocation [48] and wind-speed forecasting [49]. Furthermore, research on AOA with modifications in the exploitation phase for classification problems with high dimensional feature space [50], improved-AOA (I-AOA) [51] and enhanced-AOA (EAOA) with local escaping operator and orthogonal learning [52] for estimation of model parameters of PEM FCs are highlighting the efficiency of AOA in solving complex optimization problems and its adaptability for real-time applications.

In light of the above literature, the following are some of the most significant contributions of this research:

• (i) In this study, AOA is employed for the first time to address the DG allocation problem, and it is regarded as a key component of the entire OADG problem.

• (ii) In this work, voltage-dependent load modelling for various types of agricultural loads is taken into account.

• (iii) Loss sensitivity factors (LSFs) are recommended as a method for calculating prospective candidate sites for SPV systexms used as a search area. Following that, AOA is used to determine the best locations in the search area as well as the ideal SPV-system sizes to use.

• (iv) The results of the proposed methodology for various practical RDSs show that it is significantly more efficient in resolving the OADG problem than any other optimization algorithm in terms of minimizing actual power loss and, thus, grid dependency and GHG emissions from conventional power plants when compared to other optimization algorithms.

The remainder of the paper is organized as follows. Section 1 discusses the mathematical modelling of SPV systems and agricultural loads. Section 2 discusses the optimization problem and the associated operational constants. Additionally, methods for assessing the impact of optimal SPV-system allocations on grid dependency and GHG emissions are presented. Section 3 discusses the mathematical relationships that define the proposed AOA idea, the technique for implementing the OADG problem and the search-space reduction strategy. Section 4 of the proposed technique provides simulated results for a variety of real-world RDSs. The conclusions of this study are based on the major research findings in Section 5.

1 Mathematical modelling

This section discusses the mathematical relationships used in agriculture as well as composite load modelling, which is based on the voltage-dependent load model. To impose the influence of the SPV system on distribution networks, the appropriate power-injection modelling for the load-flow analysis of the distribution system is explained.

1.1 Voltage-dependent load modelling

In general, the voltage of the bus to which a load is connected dictates the amount of power consumed by the load. Because bus voltages in distribution networks are unpredictable, this issue must be considered. Voltage-dependent load modelling can be used to efficiently manage this issue:

$${\displaystyle \begin{array}{l}{\displaystyle P_{\left(n\right)}^{{}^{\prime }}={\displaystyle \sum _k}{\gamma }_k{P}_{\left(n\right)}{\left(\frac{{\left|V_i\right|}_{\left(t\right)}}{{\left|V_i\right|}_{\left(0\right)}}\right)}^{{\alpha }_k}}\end{array}}$$

(1)

$${\displaystyle \begin{array}{l}{\displaystyle Q_{\left(n\right)}^{{}^{\prime }}={\displaystyle \sum _k}{\gamma }_k{Q}_{\left(n\right)}{\left(\frac{\left|V_{\left(n\right)}\right|}{\left|V_{b\left(n\right)}\right|}\right)}^{{\beta }_k}}\end{array}}$$

(2)

where ${\alpha }_k$ and ${\beta }_k$ are the real and reactive power coefficients as per the type of load; ${\gamma }_k$ is the percentage of a specified type of load in the total load of the node; $P_{\left(n\right)}$ and $P_{\left(n\right)}^{\prime }$ are the nominal real power load and time-varying voltage real power loads at bus-n, respectively; $Q_{\left(n\right)}$ and $Q_{\left(n\right)}^{\prime }$ are the nominal reactive power load and time-varying voltage reactive power loads at bus-n, respectively; and $\left|V_{b\left(n\right)}\right|$ and $\left|V_{\left(n\right)}\right|$ are the nominal and time-varying voltage magnitudes of bus-n, respectively. The exponents for various load models can be found in [53, 54].

1.1.1 Constant power load

The great majority of research in the literature on the OADG problem employs a constant power load model. The real and reactive power exponents are both 0 for constant power loads in particular.

1.1.2 Agriculture load

Irrigation water-pumping systems, post-harvest and storage activities (such as parking and storage units, including dry- and cold-storage units) and processing-stage activities (such as produce drying, cereal milling, peanut value chain and edible-oil extraction, among others) account for the majority of electricity consumption in agriculture [55]. When it comes to water-pumping systems, the motor-size range varies according to the kind of irrigation, which includes drip, sprinkler and centre-pivot irrigation [56]. Room heaters and air-conditioning cold-storage units, as well as dryers and other processing equipment, are significant electricity consumers at the post-harvest stage. At each node of the test network, the following loads are distributed: 20% pumps, fans and other motors + 5% fluorescent lighting; 10% compact fluorescent lamps + 10% resistance space heater + 20% air conditioner; and a mixture of small and large industrial motors, as well as a mixture of incandescent lamps.

1.1.3 Composite load

This model assumes that the load for each bus is composed of 50% residential, 30% commercial and 20% industrial loads, respectively.

1.2 Modelling of SPVs

Solar photovoltaic electricity is often provided to the grid using DC/AC converters, which have a higher efficiency and power factor close to unity. This method can be used to calculate the actual real-time power injection from an SPV system, as well as the corresponding reactive power assistance from SPV inverters to the grid:

$${\displaystyle \begin{array}{l}{\displaystyle P_{spv(grid)}=P_{PV(c)}{\eta }_{inv}}\end{array}}$$

(3)

$${\displaystyle \begin{array}{l}{\displaystyle Q_{spv(grid)}=P_{spv(grid)}\tan \left[{\cos }^{-1}\left({\varphi }_{nv}\right)\right]}\end{array}}$$

(4)

where $P_{spv(grid)}$ and $Q_{spv(grid)}$ are the active and reactive power injections in to the grid from a SPV system, respectively; $P_{PV(c)}$ is the installed capacity of the SPV system; ${\eta }_{inv}$ and ${\varphi }_{inv}$ are the SPV inverter efficiency and its power-factor angles, respectively.

The active and reactive power injections into the grid at a load bus are modelled as negative loads at the SPV integrated bus, resulting in net-effective loads:

$${\displaystyle \begin{array}{l}{\displaystyle P_{\left(n\right)}^{{}^{\prime }}=P_{\left(n\right)}^{{}^{\prime }}-P_{spv(grid)}}\end{array}}$$

(5)

$${\displaystyle \begin{array}{l}{\displaystyle Q_{\left(n\right)}^{{}^{\prime }}=Q_{\left(n\right)}^{{}^{\prime }}-Q_{spv(grid)}}\end{array}}$$

(6)

2 Problem formulation

The goal of a solar-powered agriculture feeder is to offer electric power while reducing the reliance on demand electricity from the grid. In this context, the best SPV-system placement incorporates grid-dependency minimization and, as a result, reduced GHG emissions. This section covers the relevant objective functions as well as the operational constraints that come with them.

2.1 Multi-objective function

As described in Equations (7) and (8), the goal of this research is to minimize grid dependence by having a high PV penetration and thereby reduce GHG emissions from conventional power plants:

$${\displaystyle \begin{array}{l}{\displaystyle f_1=min\left\{G_{dp}\right\}}\end{array}}$$

(7)

$${\displaystyle \begin{array}{l}{\displaystyle f_2=min\left\{GH{G}_e\right\}}\end{array}}$$

(8)

The following constraints are considered in solving the proposed objective functions:

• bus voltage constraint: the voltage magnitude of each bus should be maintained within specified limits:

$${\displaystyle \begin{array}{l}{\displaystyle {\left|V_{\left(i\right)}\right|}_{min}\le \left|V_{\left(i\right)}\right|\le {{\left|V_{\left(i\right)}\right|}_{max}}_{}i=1,2,...,nb}\end{array}}$$

(9)

• branch thermal constraints: the current flow of any branch should not be more than its maximum rated limit:

$${\displaystyle \begin{array}{l}{\displaystyle \left|I_{\left(mn\right)}\right|\le {\left|I_{\left(mn\right)}\right|}_{max}mn=1,2,...,nl}\end{array}}$$

(10)

• DG active power compensation constraint: the total real power generation by SPV systems (⁠$P_{spv\left(T\right)}$⁠) should not be more than the total effective real power load (⁠$P_{L\left(T\right)}^{\prime }$⁠) in the network:

$${\displaystyle \begin{array}{l}{\displaystyle P_{spv\left(T\right)}\le P_{L\left(T\right)}^{{}^{\prime }}}\end{array}}$$

(11)

• DG reactive power compensation constraint: the total reactive power compensation via SPV inverters should not be more than the total effective reactive power load of in the network:

$${\displaystyle \begin{array}{l}{\displaystyle Q_{inv\left(T\right)}\le Q_{L\left(T\right)}^{{}^{\prime }}}\end{array}}$$

(12)

2.2 Computation of grid dependency and GHG emissions

Depending on the extent of penetration, the net-effective loading of the feeder may decrease as a result of optimizing the capacity of SPV systems. The percentage of grid dependency of the network is calculated using the optimization findings (defined as the ratio of the total power drawn from the grid after SPV allocation to the total power drawn from the grid before SPV integration):

$${\displaystyle \begin{array}{l}{\displaystyle G_{dep}=\left\{\frac{real{\left(S_{\left(s/s\right)}\right)}_{spv}}{real{\left(S_{\left(s/s\right)}\right)}_{base}}\right\}\times 100\mathrm{\%}}\end{array}}$$

(13)

CO₂, SO₂ and NO_x are the most hazardous pollutants produced by conventional power plants in terms of GHG emissions. It is feasible to reduce real power loss and dependency on grid power by allocating SPV systems optimally, reducing the quantity of electricity generated from the grid and, as a result, reaping environmental benefits. The emission factors for NO_x, SO₂ and CO₂ using grid electricity are 2.2952, 5.2617 and 921.2461 kg/MWh, respectively [57].

The total power generation, which includes the whole real power demand as well as distribution losses, can be used to compute the emissions from conventional power plants. This is simply the entire amount of real electricity drawn from the grid at the substation bus. As a result, GHG emissions are represented by:

$${\displaystyle \begin{array}{l}{\displaystyle E_{m\left(grid\right)}=(C{O}_2+N{O}_x+S{O}_2)\times real\left(S_{\left(s/s\right)}\right)}\end{array}}$$

(14)

Because the SPV systems directly reduce the grid supply to the network, the total amount of GHG emissions is reduced by a significant amount as well.

2.3 Assessment of distribution system losses

The backward/forward load-flow [58] solution is used to determine the voltage profile and consequently the overall performance of the network. Let us consider a branch between buses m and n, having an impedance of $z_{\left(mn\right)}=r_{\left(mn\right)}+j{x}_{\left(mn\right)}$⁠. And, the complex power connected at the receiving end bus-n is $S_{\left(n\right)}=P_{\left(n\right)}^{\prime }+j{Q}_{\left(n\right)}^{\prime }$⁠. Now, the receiving end bus-n voltage can be determined by:

$${\displaystyle \begin{array}{l}{\displaystyle \left|V_{\left(n\right)}\right|={(k_2-k_1)}^{\frac{1}{2}}}\end{array}}$$

(15)

$${\displaystyle \begin{array}{l}{\displaystyle k_1=r_{\left(mn\right)}{P}_{\left(n\right)}^{{}^{\prime }}+x_{\left(mn\right)}{Q}_{\left(n\right)}^{{}^{\prime }}-0.5{\left|V_{\left(m\right)}\right|}^{2}and{k}_2=A^2-{\left(z_{\left(mn\right)}^2{S}_{\left(n\right)}^2\right)}^{\frac{1}{2}}}\end{array}}$$

(16)

Now, the real and reactive power losses can be determined by:

$${\displaystyle \begin{array}{l}{\displaystyle P_{loss,mn}={\left|I_{\left(mn\right)}\right|}^2{r}_{\left(mn\right)}=\left(\frac{P_{\left(n\right)}^{{}^{\prime }2}+Q_{\left(n\right)}^{{}^{\prime }2}}{{\left|V_{\left(n\right)}\right|}^2}\right)r_{\left(mn\right)}}\end{array}}$$

(17)

$${\displaystyle \begin{array}{l}{\displaystyle Q_{loss,mn}={\left|I_{\left(mn\right)}\right|}^2{x}_{\left(mn\right)}=\left(\frac{P_{\left(n\right)}^{{}^{\prime }2}+Q_{\left(n\right)}^{{}^{\prime }2}}{{\left|V_{\left(n\right)}\right|}^2}\right)x_{\left(mn\right)}}\end{array}}$$

(18)

n

$${\displaystyle \begin{array}{l}{\displaystyle P_{\left(m\right)}^{{}^{\prime }}=P_{\left(m\right)}^{{}^{\prime }}+P_{loss,mn}and{Q}_{\left(m\right)}^{{}^{\prime }}=Q_{\left(m\right)}^{{}^{\prime }}+Q_{loss,mn}}\end{array}}$$

(19)

This sequence continues for each lateral in a backward process up to the substation bus until the load-flow convergence criterion is satisfied. Hence, the total distribution losses (⁠$S_{oss\left(T\right)}$⁠) are equal to the difference between the net-effective loading at the substation bus (⁠$S_{\left(s/s\right)}$⁠) and the total load (⁠$S_{L\left(T\right)}$⁠) of the network, and is given by:

$${\displaystyle \begin{array}{l}{\displaystyle S_{loss\left(T\right)}=S_{\left(s/s\right)}-S_{L\left(T\right)}}\end{array}}$$

(20)

Equation (19) demonstrates that branch currents and losses are directly proportional to the total connected load at each bus in the network, and that by employing dynamic power compensation, as defined by Equations (5) and (6), the net-effective loading at specific locations can be reduced, and as a result, the branch currents and losses can be reduced as well. Consequently, the identification of appropriate buses as well as values for DG sizes constitutes the most significant search space in the OADG problem. As a result, discrete and continuous variables should be tuned for the location and size of DGs in the search-variable tuning process.

The randomly generated variables (as well as the positions and sizes of DGs) within the given limits are used as inputs for each iteration of the optimization process to adjust the data in the test system. There, the net-effective loading is applied using randomly generated DG powers and a load flow is produced to calculate the total losses of the network. To put it another way, each iteration generates random DG locations and sizes, and this procedure is repeated until the proposed algorithm satisfies the convergence condition.

2.4 Assessment of average voltage deviation

Given the importance of voltage quality in determining the reliability of service in any EDN, calculating the average voltage deviation (AVD) of all buses in relation to the substation voltage can be a valuable indicator. As a result, when creating a multi-objective function, the minimization of AVD is also considered, and the resulting function is as follows:

$${\displaystyle \begin{array}{l}{\displaystyle AVD=\frac{1}{nb}{\left[{\displaystyle \underset{i=1}{{\displaystyle \sum ^{nb}}}}{(\left|V_{\left(s/s\right)}\right|-\left|V_{\left(i\right)}\right|)}^2\right]}^{\frac{1}{2}}}\end{array}}$$

(21)

2.5 Assessment of the voltage stability index (VSI)

Maintaining an adequate voltage-stability margin at the distribution system level is important to averting the collapse of the entire power system, as has already occurred [59]. Numerous indices for evaluating the voltage stability of RDSs have been developed in the literature [60]. In this investigation, it was chosen to adopt the VSI proposed in [61], which is defined as follows:

$${\displaystyle \begin{array}{l}{\displaystyle VSI={\left|V_{\left(m\right)}\right|}^4-4(x_{\left(mn\right)}{P}_{\left(n\right)}^{{}^{\prime }}-r_{\left(mn\right)}{Q}_{\left(n\right)}^{{}^{\prime }})-4(r_{\left(mn\right)}{P}_{\left(n\right)}^{{}^{\prime }}+x_{\left(mn\right)}{Q}_{\left(n\right)}^{{}^{\prime }}){\left|V_{\left(n\right)}\right|}^2}\end{array}}$$

(22)

where m is the sending end bus, and n is the receiving end bus of any line; V_(m) and V_(n) are the voltages at bus-m and bus-n, respectively; P’_(n) and Q’_(n) are the real and reactive powers at the nth bus, respectively; r_(mn) and x_(mn) re the resistance and reactance of the line mn. The nearer the value of index is to one, the more stable the system is.

3 Archimedes optimization algorithm

According to Archimedes’ principles, the upward buoyant force that is exerted on an item immersed, whether partially or fully submerged in a liquid, is equal to the weight of the liquid that the item displaces and acts in the upward direction at the centre of mass of the displaced liquid [62]. By mimicking this principle of physics, a new population-based metaheuristic technique called AOA was introduced by Hashim et al. in 2021 [44]. Similar to all other population-based algorithms, AOA also followed the initialization phase for generating a random population, evaluation of the population and updating parameters in the exploration and exploitation phases and termination phase. In this section, the detailed mathematic relations of AOA are explained as per the steps to be followed while solving an optimization problem.

3.1 Algorithmic phases

3.1.1 Initialization phase

In the initialization phase, each item is considered to be positioned randomly in the liquid. The items are treated as population or search variables, and their positions are randomly generated within their lower and upper limits. By treating the number of items as a dimension of search variables $d_s$⁠, the initial population is generated by:

$${\displaystyle \begin{array}{l}{\displaystyle X_k=rand\cdot \ast [U_k-L_k]+L_k;k=1:d_s}\end{array}}$$

(23)

where $X_k$ is the kth item in a population of $d_s$ items; and $L_k$ and $U_k$ are the lower and upper boundaries of the search variables, respectively.

At this stage, the volume (⁠$V_k$⁠), density (⁠$D_k$⁠) and acceleration (⁠$A_k$⁠) for each kth item are initialized by using $r_k$⁠, which is a $d_s$ dimensional vector randomized in the range of [0, 1] and given by:

$${\displaystyle \begin{array}{l}{\displaystyle V_k=r_k;D_k=r_k\mathrm{a}\mathrm{n}\mathrm{d}{A}_k=rand\cdot \ast [U_k-L_k]+L_k}\end{array}}$$

(24)

Finally, the best item and its fitness value are finalized by evaluating the fitness function in this phase—let them be denoted as $X_{k,best}$⁠, $A_{k,best}$⁠, $D_{k,best}$ and $V_{k,best}$ for best item k and its best acceleration, best density and best volume, respectively.

3.1.2 Updating variables phase

In this phase, the density and volume of the kth item for the next iteration $(it+1)$ is updated by:

$${\displaystyle \begin{array}{l}{\displaystyle D_k^{it+1}=D_k^{it}+r_k[D_{k,best}-D_k^{it}]}\end{array}}$$

(25)

$${\displaystyle \begin{array}{l}{\displaystyle V_k^{it+1}=V_k^{it}+r_k[V_{k,best}-V_k^{it}]}\end{array}}$$

(26)

where $D_{k,best}$ and $V_{k,best}$ are the best density and volume of best item k evaluated so far, respectively; and its best acceleration, best density and best volume, respectively; $r_k$ is a random number with uniform distribution.

3.1.3 Generation of adaptive parameters

Specifically, the computational efficiency and the capability of escaping from a local optima trap of any population algorithm is mainly dependent upon its control parameters. In AOA also, two such parameters are defined for balancing between exploration and exploitation phases, namely the transfer operator (TO) and density factor (DF). The TO is used to mimic the initial collision stage of items and then their stabilization after some time, thus it is modelled as exponentially increasing until it reaches 1 and is given by:

$${\displaystyle \begin{array}{l}{\displaystyle TO=\exp \left(\frac{it-i{t}_{max}}{i{t}_{max}}\right)}\end{array}}$$

(27)

where $it$ and $i{t}_{max}$ are the current iteration and maximum number of iterations, respectively.

Similarly, the decreasing density nature of the item as time progresses is represented by using the DF for arriving at the global optima and is modelled as:

$${\displaystyle \begin{array}{l}{\displaystyle D{F}^{it+1}=\exp \left(\frac{i{t}_{max}-it}{i{t}_{max}}\right)-\left(\frac{it}{i{t}_{max}}\right)}\end{array}}$$

(28)

3.1.4 Exploration phase

This phase is proposed when collisions occur between items. It is assumed that the collision of items takes place for $TO\le 0.5$ and correspondingly such items experience acceleration, as updated by using a random material (⁠$mr$⁠):

$${\displaystyle \begin{array}{l}{\displaystyle {A_k}^{it+1}=\frac{D_{mr}+V_{mr}{A}_{\mathrm{m}r}}{{D_k}^{it+1}+{V_k}^{it+1}}}\end{array}}$$

(29)

where $D_{mr}$⁠, $V_{mr}$ and $A_{mr}$⁠, and $D_k$⁠, $V_k$ and $A_k$ are the density, volume and acceleration of the random material/item and the kth item, respectively. Here, $TO\le 0.5$ is expedited for completing one-third of the total number of iterations; by changing this, the exploration–exploitation behaviour of AOA can be changed.

3.1.5 Exploitation phase

This phase is proposed when items are not subjected to any collision, i.e. $TO>0.5$⁠, and thus the acceleration of the kth item is updated by using the best values of density, volume and acceleration obtained so far:

$${\displaystyle \begin{array}{l}{\displaystyle A_k^{it+1}=\frac{D_{k,best}+V_{k,best}{A}_{k,best}}{{D_k}^{it+1}{V_k}^{it+1}}}\end{array}}$$

(30)

3.1.6 Transformation from exploration to exploitation phase

In this phase, the percentage of step change that each item gains in the exploration and exploitation phases is modelled in the range of [$L$⁠, $U$] and is given by:

$${\displaystyle \begin{array}{l}{\displaystyle A_{k(norm)}^{it+1}=U\left(\frac{A_k^{it+1}-A_{min}}{A_{max}-A_{min}}\right)+L}\end{array}}$$

(31)

The lower and upper ranges of normalization are set to 0.1 and 0.9, respectively. The items are usually in the exploration phase when $A_{k(norm)}^{it+1}$ is a high value, representing a position far away from the global optima; otherwise, they exist in the exploitation phase. As iterations progress, $A_{k(norm)}^{it+1}$ spans from high values to low values and ensures that the search space moves towards the global optima.

3.1.7 Updating variables position phase

The position of the kth object for the next iteration it + 1 is updated using Equation (32) if TO ≤ 0.5 or else using Equation (33) if TO ≥ 0.5:

$${\displaystyle \begin{array}{l}{\displaystyle X_k^{it+1}=X_k^{it}+K_1{r}_k{A}_{k(norm)}^{it+1}D{F}^{it+1}(T{X}_{best}-X_k^{it});TO\le 0.5}\end{array}}$$

(32)

$${\displaystyle \begin{array}{l}{\displaystyle X_k^{it+1}=X_{k,best}^{it}+F_k{K}_2{r}_k{A}_{k(norm)}^{it+1}D{F}^{it+1}(T{X}_{best}-X_k^{it});TO>0.5}\end{array}}$$

(33)

$${\displaystyle \begin{array}{l}{\displaystyle F_k=\{\begin{array}{l}\begin{array}{lll}+1& {\displaystyle if}& {\displaystyle (P=2r_k-K_4)\le 0.5}\end{array}\\ \begin{array}{lll}-1& {\displaystyle if}& {\displaystyle (P=2r_k-K_4)>0.5}\end{array}\end{array}}\end{array}}$$

(34)

where $T=K_{3}TO$ is directly proportional to TO that lies between [$0.3K_3$⁠, 1]; $F$ is a flag used for change the direction of the movement of items; $K_1$ and $K_2$ are constants equal to 2 and 6, respectively; $K_3$ and $K_4$ are also constants set as 2 and 0.5, respectively, for engineering optimization problems, otherwise they are set to 1 and 4 for standard optimization functions.

Finally, the global best item and its fitness value are finalized by evaluating the fitness function in this phase. Once the iteration count reaches the maximum, $it=i{t}_{max}$⁠, then the optimization process is complete. The overall methodology is given as a flowchart in Fig. 1.

Fig. 1:

Flowchart of AOA implementation procedure for solving DG allocation problem.

3.2 Methodology for locations and SPV sizes equal to penetration level

In this section, the methodology followed to identify potential candidates for integrating SPV systems and the application of AOA to solve OADG problems is explained. By this, the search space of SPV systems can be reduced only for potential candidate locations, thus AOA can exhibit fast convergence characteristics.

3.2.1 Identification of potential candidates for SPV systems

Loss minimization is one of the aspects in the minimization of grid dependency. Hence, by having real power compensation by a SPV system at a specified bus, the real power loss sensitivity indexes (⁠$LS{I}_p$⁠) are expressed.

The real power loss of a branch in a distribution system is given by:

$${\displaystyle \begin{array}{l}{\displaystyle P_{loss\left(mn\right)}=r_{\left(mn\right)}\left(\frac{P_{\left(n\right)}^{{}^{\prime }2}+Q_{\left(n\right)}^{{}^{\prime }2}}{{\left|V_{\left(n\right)}\right|}^2}\right)}\end{array}}$$

(35)

The $LS{I}_p$ is defined as the change in real power loss of a branch with respect to change in load at its receiving bus, mathematically:

$${\displaystyle \begin{array}{l}{\displaystyle {\mathrm{L}\mathrm{S}\mathrm{I}{}_{\mathrm{p}}}_{\left(\mathrm{n}\right)}=\frac{\mathrm{\partial }{P}_{loss\left(mn\right)}}{\mathrm{\partial }{P}_{\left(n\right)}^{{}^{\prime }}}=\frac{2r_{mn}{P}_{\left(n\right)}^{{}^{\prime }}}{{\left|V_n\right|}^2}}\end{array}}$$

(36)

where LSI(p) is the voltage sensitivity index of bus n.

Using the base-case load-flow result, $LS{I}_p$ is determined for all the buses using Equation (36). Later, by sorting the buses into descending order as per the $LS{I}_p$ values, the top 10 ordered locations are considered as a predefined input array (⁠$L_{pre}$⁠) for the SPV locations:

$${\displaystyle \begin{array}{l}{\displaystyle {\mathrm{L}\mathrm{S}\mathrm{I}}_{(\mathrm{p})}=sort(\left[LS{F}_i\right],{}^{\prime }descend{}^{\prime })}\end{array}}$$

(37)

Hence, the search variables for SPV locations (⁠$X_{loc}$⁠) are generated in the range [1, 10]. Since the generated random numbers are in continuous form, the function $randi()$ is used for transforming them into discrete variables (⁠$L_{AEO}$⁠). Now, the corresponding SPV location (⁠$L_{Final,i}$⁠) is picked from the predefined search space:

$${\displaystyle \begin{array}{l}{\displaystyle L_{AEO}=randi\left(X_{loc}\right)}\end{array}}$$

(38)

$${\displaystyle \begin{array}{l}{\displaystyle L_{Final,i}=L_{pre}\left(L_{AEO,i}\right)}\end{array}}$$

(39)

This stage can significantly minimize the computational time of AOA with an iteration. And, it can also ensure the integration of SPV locations only at high potential candidates from a loss-reduction point of view.

3.2.2 Generation of sizes of SPV systems equal to required penetration level

In this work, the penetration level of SPV systems is modelled as a fraction of the total load of the network. Hence, the SPV sizes are generated using the following random theory. Initially, an array of non-zero random numbers (⁠$A$⁠) are generated as per the number of SPV systems (⁠$n_{spv}$⁠) to be integrated in the network and later a revised array (⁠$B$⁠) is obtained as:

$${\displaystyle \begin{array}{l}{\displaystyle A=rand(1,n_{spv}),B=A/sum\left(A\right)}\end{array}}$$

(40)

Now, by multiplying the revised array (⁠$B$⁠) by the total load, it is possible to get the sum of all SPV-system sizes as equal to the total load. By choosing a scaling factor (⁠${\lambda }_{pen}$⁠) between (0, 1), the required penetration level of SPV systems can be obtained in the network:

$${\displaystyle \begin{array}{l}{\displaystyle P_{spv(grid)\left(i\right)}=B_i{P}_{L\left(T\right)}^{{}^{\prime }}{\lambda }_{pen},i=\mathrm{1...}{n}_{spv}}\end{array}}$$

(41)

This simple approach can reduce the computational burden for ensuring that the total SPV-system sizes are equal to the required penetration level. In the existing literature, the optimal sizes of SPV systems can be generated, but they are not suited for generating optimal sizes of SPV systems at a desired penetration level.

4 Results and discussion

The simulations are done on a PC with 4 GB of RAM, a 64-bit operating system and an Intel® CoreTM i5-2410M CPU running at 2.30 GHz, all with the help of MATLAB. Different case studies are performed on three practical distribution feeders. The first feeder is a 22-bus feeder that represents a small portion of the agricultural distribution in Warangal rural region in the state of Telangana, India [63]. The second feeder is a 28-bus feeder that represents a rural region of Kakdwip in the state of West Bengal, India [64]. The third feeder is a 85-bus feeder that represents an urban area of Mysore City in the state of Karnataka, India [65]. The maximum number of iterations and population in all case studies is set to 50. In the first phase, the AOA results are provided based on technological benefits and then the environmental benefits are mentioned.

4.1 Discussion considering technical aspects

4.1.1 Practical 22-bus agriculture feeder

The feeder has 22 buses coupled by 21 branches as depicted in Fig. 2 and runs at an 11-kV voltage. The constant power load model has been adjusted to more closely reflect the various types of loads found in the agricultural sector, as previously described in Section 1.1.2. As a result, the effective loading of the feeder is altered to (655.58 kW + j 646.4794 kVAr), while the distribution losses are modified to (16.9844 kW + j 8.6911 kVAr). Furthermore, the 22nd bus has the lowest voltage magnitude, AVD and VSI for the entire system, with values of 0.9783 p.u., 0.027 and 0.9162, respectively. The base case refers to this level of performance.

Fig. 2:

Single-line diagram of practical 22-bus agriculture feeder.

The overall grid dependency of the network for real power consumption is ~663.92 kW (including load and losses) and it is intended that, by allocating SPV systems in the most efficient manner, this dependency can be reduced. In this network, only loss minimization (F₁), as specified in Equation (7), is considered the primary goal for minimization. The proposed AOA is utilized to most efficiently interconnect three SPV systems. As a result, the number of search results has been increased to six (i.e. three for locations and three for sizes). In SPV-system design, the operating power factor and efficiency of the grid-connected inverter are considered to be 0.98 lagging and 95%, respectively.

The LSI values are used to determine the predefined search-space SPV locations. Based on the base-case solution, the buses are graded and depicted in Fig. 3. From among the top 10 selected places, AOA is now in charge of determining the most optimum locations. Furthermore, the total load of the system, which is 655.58 kW (i.e. 100% SPV_IC level), must equal the sum of the optimized SPV capacities (SPV_IC), resulting in the unequal constraint expressed by Equation (11) becoming an equal constraint to the optimization problem, and so the optimization problem is solved.

Fig. 3:

Ranking of buses as per LSI values in 22-bus EDN.

Among the 25 simulation trails tested, the following is the best AOA solution: for buses 2, 14 and 19, the best SPV-system sizes in kW are 123.025, 305.895 and 227.079, respectively. As a result of the real power adjustment supplied by SPV systems, overall losses are decreased to (4.374 kW + j 2.238 kVAr) and the effective loading of the system is equal to (38.911 kW + j 528.847 kVAr). The lowest voltage is raised to 0.9954 p.u. on the ninth bus, the maximum voltage is raised to 0.9819 and the lowest current is equal to 0.0007. When compared to the base-case configuration, the grid dependency of the system for real power has been lowered by 6.45% as a result of this optimal allocation of SPV systems with a total installed capacity of 655.28 kW. The voltage profile is depicted in Fig. 4 without and with SPV-system allocation, respectively. The appropriate allocation of SPV systems, it can be argued, resulted in a significant improvement in the voltage profile as a result of the optimization. Furthermore, the voltage profile of the network has become flatter, which is crucial for long-term stability. The VSI has increased in value, indicating that the voltage stability of the network has improved.

Fig. 4:

Voltage profile of 2-bus system with SPV systems.

Particle swarm optimization (PSO) [66], ALO [67], grasshopper optimization (GHO) [68], teaching–learning-based optimization (TLBO) [69] and the cuckoo search algorithm (CSA) [70] are used for the comparison of AOA performance while solving SPV allocation for a 100% penetration level. The common controlling parameters are as follows: population size = 50, maximum iterations = 50, number of variables = 6 and number of independent runs = 25; other specific controlling parameters of each algorithm are as follows: for PSO, $c_1$= 2, $c_2$= 2 and linearly decreases inertia weight $w_i$= [0.9, 0.4]; for GHO, minimum and maximum values for $c_1$ and $c_2$ are 0.00004 and 1, respectively; for TLBO, teaching factor (T_F) = 1 or 2; and for CSA, switching parameter $P_a$= 0.25; for AOA, $K_1$= 2, $K_2$= 6, $K_3$= 2 and $K_4$= 0.5, respectively. Table 1 shows the best result from those simulations. According to the various performance indices shown in Table 2, AOA outperforms other algorithms, with lower average and standard deviation values emphasizing the superiority of AOA over other algorithms with lower average and standard deviation values. Additionally, as compared to other methods, AOA takes less time to compute. The convergence characteristics of these algorithms are given in Fig. 5.

Table 1:

Results of 22-bus with agriculture loads by different algorithms

Algorithm	SPV Locations			SPV Sizes (kW)			Ploss (kW)	Qloss (kW)	Vmin (p.u.)	AVD	VSI	Griddep (%)
Base	-			-			14.020	7.174	0.9821 (22)	0.0027	0.9304	100
PSO	19	2	6	316.27	216.42	123.31	5.78	2.96	0.992 (18)	0.0012	0.9699	6.60
TLBO	19	6	2	316.28	123.31	216.42	5.78	2.96	0.992 (18)	0.0012	0.9699	6.60
CSA	13	14	2	237.94	239.75	178.31	5.32	2.72	0.991 (22)	0.0012	0.9653	6.50
ALO	19	2	16	331.67	192.72	131.61	4.75	2.43	0.995 (8)	0.0009	0.9783	6.48
GOA	2	17	16	162.91	221.95	271.14	4.63	2.37	0.994 (22)	0.0009	0.9762	6.47
AOA	2	14	19	123.03	305.90	227.08	4.37	2.24	0.995 (8)	0.0007	0.9819	6.45

Table 2:

Performance indices of different algorithms for 22-bus with agriculture loads

Performance Index	Algorithm
	PSO	ALO	GHO	TLBO	CSA	AOA
Min	5.7760	5.7760	5.3180	4.7450	4.6320	4.3740
Max	5.7996	5.8477	5.3381	4.7661	4.6692	4.3929
Avg	5.7810	5.7792	5.3224	4.7477	4.6356	4.3758
Std	0.0062	0.0051	0.0052	0.0052	0.0062	0.0046
Avg. Time (sec)	30.4247	30.2713	29.9826	29.9645	29.8726	29.8243

Fig. 5:

Convergence characteristics of different algorithms.

4.1.2 Practical 28-bus rural feeder

As illustrated in Fig. 6, the test system consists of 28 buses connected by 27 branches. The overall operating power factor of the feeder is 0.7, which is underperforming. As mentioned in Section 1.1.3, the load on each bus is assumed to be 50% residential, 30% commercial and 20% industrial, with the remaining 20% being a combination of the three. Following the execution of the load flow, it is revealed that the effective loading of the system is (715.39 kW + j 599.81 kVAr), with resultant losses of (48.3684 kW + j 32.4271 kVAr). Furthermore, the 26th bus has the lowest voltage magnitude of 0.9258 p.u. and a voltage sensitivity index of 0.7348.

Fig. 6:

Single-line diagram of particle 28-bus agriculture feeder.

The optimal allocation of SPV systems can be solved for Scenario 1, which is the minimizing of only losses (i.e. F₁) at the 100% SPV_IC level; and Scenario 2, which is the minimization of both losses and AVD at the 100% SPV_IC level (i.e. F₂).

Scenario 1: Optimal allocation of SPV systems for minimization of only losses considering 100% SPV_IC level

This example shows how AOA may be utilized to minimize losses by integrating three SPV systems optimally. Bus numbers 22, 4, 6, 23, 12, 24, 7, 3, 27 and 11 have been recognized as potential candidates for SPV locations by local service providers. The remaining input variables are the same as in the 22-bus feeder before it. Furthermore, the total load of the system, which is 715.39 kW, must be equal to the sum of the optimal SPV capacities. Among the 25 simulation trails tested, the following is the best AOA solution.

The power outputs of the best SPV systems in kW are 315.48 (6), 190.73 (12) and 209.18 (24). Total losses were cut in half to (18.036 kW + j 11.926 kVAr)—a substantial reduction. The system may run at an effective loading of (75.7 kW + 577.85 kVAr), which is equal to (75.70 kW + 577.85 kVAr) thanks to real power correction. The lowest voltage is raised to 0.9712 p.u., the greatest voltage, on the 28th bus. The grid dependency for actual power is 11.58%, which is lower than the grid dependency for uncompensated power, according to the optimal allocation of SPV systems with a total installed capacity of 715.39 kW. Table 3, which also includes results from the AOA research, has results from different heuristic search algorithms (HSAs). AOA, as in prior testing, beat other algorithms in terms of providing the least objective function in this test system.

Table 3:

Results of 28-bus rural feeder with agriculture loads by different algorithms

Algorithm	SPV Locations			SPV Sizes (kW)			Ploss (kW)	Qloss (kW)	Vmin (p.u.)	AVD	VSI	Griddep (%)
PSO	24	12	8	200.95	255.58	258.86	18.15	11.98	0.971 (28)	0.0040	0.8897	14.02
TLBO	24	12	6	248.46	209.28	257.65	18.14	11.94	0.971 (28)	0.0042	0.8899	13.98
CSA	6	22	12	272.01	292.39	150.99	18.13	11.90	0.974 (28)	0.0041	0.8988	14.02
ALO	22	6	12	277.32	220.74	217.33	18.10	11.90	0.972 (28)	0.0043	0.8910	13.97
GHO	6	24	12	317.54	200.68	197.16	18.04	11.94	0.971 (28)	0.0043	0.8885	13.97
AOA	6	12	24	315.48	190.73	209.18	18.04	11.93	0.971 (28)	0.0043	0.8896	13.97

Scenario 2: Minimization of both losses and AVD (i.e. OF₂) considering different SPV_IC levels

In this case, AOA is utilized to solve a multi-objective function for the simultaneous minimization of both losses and AVD at various SPV penetration levels, with the results displayed in a graph. At 100% SPV_IC, the SPV systems with the highest power production are 261.456 kW, 194.451 kW, 259.483 kW at buses 6, 11 and 22, respectively (6). (22). The effective loading of the system is (75.564 kW + j 576.893 kVAr) as a result of the real power adjustment, and the grid dependency of the system for real power is ~12.27% lower than in the base scenario, as indicated in Fig. 7. The total losses were reduced by a significant amount to (18.1733 kW + j 11.9212 kVAr). The lowest voltage is raised to 0.9719 p.u. on the 28th bus, which is the highest voltage ever recorded. Table 4 demonstrates the best EDN-node placements and sizes for different SPVIC levels, as well as EDN-node performance. Aside from that, Fig. 7 shows the voltage patterns for varied penetration levels. The results show that network performance increases in direct proportion to the penetration level, and that the reliance of the network on the grid is greatly decreased.

Table 4:

Results of AOA for minimization of both losses and AVD simultaneously for different SPV_IC levels (28-bus, composite loads)

SPVIC level	SPV Locations			SPV Sizes (kW)			Ploss (kW)	Qloss (kW)	Vmin (p.u.)	AVD	VSI	Griddep (%)
Base							14.02	7.17	0.982 (22)	0.0027	0.9304	100
50%	11	7	24	118.012	172.451	67.232	26.5775	17.7299	0.9507 (26)	0.0021	0.8168	56.15
60%	7	24	11	206.938	80.660	141.636	23.8235	15.8653	0.9556 (26)	0.0020	0.8341	47.38
70%	11	6	22	136.116	183.019	181.638	21.9064	14.5053	0.9586 (26)	0.0018	0.8443	38.59
80%	7	11	24	275.961	188.799	107.553	19.9114	13.2053	0.9641 (21)	0.0018	0.8641	29.84
90%	6	24	11	282.746	186.100	175.005	18.9102	12.5035	0.9668 (28)	0.0016	0.8735	21.04
100%	6	11	22	261.456	194.451	259.483	18.1733	11.9212	0.9719 (28)	0.0015	0.8924	12.27

Fig. 7:

Voltage profile in 28-bus system for penetration levels of different SPV systems.

4.1.3 Practical 85-bus urban feeder

Fig. 8 shows an 85-bus feeder with a single bus in a single-line diagram. The system is made up of 84 branches that connect 85 buses to supply the total load of (2570.28 kW + j 2622.08 kVAr). The working voltage of the feeder is 11 kV and load flow reveals that the overall distribution losses are (315.926 kW + j 198.483 kVAr), with the lowest voltage at the 54th bus, being 0.8717 p.u. (315.926 kW + j 198.483 kVAr) are the overall distribution losses.

Fig. 8:

Single-line diagram of particle 85-bus urban feeder.

There are three scenarios for this test system, namely Scenario 1 (standard constant power loads), Scenario 2 (combined load modelling considering residential, industrial and commercial-type consumers) and Scenario 3 (combined load modelling considering agricultural loads).

Scenario 1: Constant power loads

The top 10 candidate sites (buses 8, 6, 7, 58, 4, 27, 25, 3, 29 and 34) are selected as the search space for a specific place, according to the LSFs. The findings of the AOA at each penetration depth are shown in Table 5. When we look at the data, we can see that as penetration increases, losses reduce and the dependency of the grid on the system drops considerably.

Table 5:

Results of AOA for different SPV_IC levels in 85-bus system with constant power loads

SPVIC level	Locations			SPV Sizes			Ploss (kW)	Qloss (kW)	Vmin (p.u.)	Griddep (%)
50%	58	58	34	123.99	359.64	801.51	146.09	91.55	0.9287 (76)	53.78
60%	58	34	8	771.08	771.08	0.00	129.37	80.47	0.9384 (76)	44.46
70%	58	27	34	792.49	386.81	619.90	116.32	71.89	0.9450 (76)	35.20
80%	58	29	34	928.49	390.64	737.10	107.55	65.81	0.9527 (76)	26.05
90%	34	58	8	815.26	827.19	670.80	102.30	62.22	0.9577 (76)	17.04
100%	8	34	58	1021.85	803.75	744.68	99.30	60.24	0.9629 (76)	8.14

When loss minimization (i.e. OF₁) is taken into account for different SPV_IC levels, Table 5 indicates the best position and size of SPV systems, as well as the related performance of the test system (i.e. OF₁). For 100% SPV_IC, the best SPV systems are 1021.85 kW, 803.75 kW, 744.68 kW at buses 8, 34 and 58, respectively. (58). Total losses were reduced by a large amount to (99.302 kW + j 60.236kVAr). The lowest voltage is raised to 0.9629 p.u., which is the greatest voltage, on the 76th bus. The grid dependency of the system for real power is 8.142%, which is lower than the uncompensated system when comparing the optimal allocation of SPV systems with a total installed capacity of 2570.28 kW. For different penetration levels, the voltage profile is given in Fig. 9. By repeating the simulations with a 100% SPV_IC level utilizing the aforementioned HSAs, the results of which can be found in Table 6, AOA is compared to other HSAs.

Table 6:

Comparison of different algorithms for 100% SPV_IC level in 85-bus system with constant power loads

Algorithm	Locations			SPV Sizes			Ploss (kW)	Qloss (kW)	Vmin (p.u.)	Griddep (%)
Base	-			-
PSO	7	58	34	393.00	1184.93	992.34	102.620	61.805	0.9632 (76)	8.251
TLBO	27	34	58	332.54	905.97	1331.77	101.148	60.753	0.9692 (76)	8.202
CSA	27	34	58	760.61	525.64	1284.03	100.692	60.648	0.9687 (76)	8.187
ALO	8	58	34	1211.71	547.09	811.48	100.236	60.712	0.9608 (76)	8.172
GHO	34	58	27	605.22	1158.87	806.20	100.132	60.397	0.9674 (76)	8.169
AOA	8	34	58	1021.85	803.75	744.68	99.302	60.236	0.9629 (76)	8.142

Fig. 9:

Voltage profile in 85-bus system for different SPVIC levels considering constant power loads.

Table 7 compares and contrasts further literature works on OADG difficulties in 85-bus systems (i.e. SPV systems with 100% efficient inverters and unity power factor). The overall capacity of the SPV systems is less than the entire connected load in this comparison, resulting in a heavy reliance on the power grid. In this scenario, AOA is likewise found to be the most complete.

Table 7:

Comparison of literature in 85-bus system with constant power loads

Reference	Locations			SPV Sizes(kW)			Ploss (kW)	Griddep (%)
MRFO [8]	11	64	34	617.15	654.89	732.85	149.348	24.76
APSO [11]	9	34	67	856.67	731.94	607.16	148.271	18.11
MGSA [11]	9	34	65	790.65	766.86	662.06	149.217	17.32
GA [14]	30	60	49	630.00	940.00	210.00	155.876	32.78
SA [14]	63	26	47	830.00	820.00	340.00	156.066	25.51
PSO [14]	33	14	73	1000.00	450.00	560.00	153.827	24.74
AQiEA [14]	66	26	49	620.00	1000.00	455.80	151.964	22.40
SSA [15]	8	33	66	1068.00	701.00	500.00	150.362	15.65
HGWO [25]	48	70	9	416.00	198.00	1876.00	154.089	8.12
AOA	9	34	67	1100.00	660.00	550.00	148.123	14.15

Scenario 2: Agricultural loads

As indicated in Section 1.1.3, the characteristics of the feeder constant power load are changed into those of agricultural loads in this situation. The simulations are run only for the aim of minimizing losses (i.e. OF₁). Because the voltage profile and voltage stability of a network are directly proportional to the amount of reactive power reserve available, it is also important to look into how SPV systems help with reactive power adjustment by running their inverters at ideal power factors. As a result, the operational power factors of the inverters are optimized in this case study and the operating range is estimated to be between [0.8 and 1.0]. The efficiency of the inverter is estimated to be 95%. As a result, the search space of the optimization process expands to nine (i.e. three for locations, three for sizes and three for power factors).

When voltage-dependent load modelling is applied to compute the load flow, the overall agriculture loading effect is equal to (2442.55 kW + j 2407.36 kVAr) and the total loss is equal to (264.011 kW + j 166.095 kVAr). The bus with the lowest voltage, 0.8825 p.u., is found on the 54th bus of the feeder. As a result, the AVD and VSI in this investigation were found to be 0.00976 and 0.6066, respectively.

For a 100% SPV_IC level, the optimal SPV systems in kW are 592.83 (51), 1070.65 (11) and 779.07 kW (66), respectively. Total losses were decreased to (39.70 kW + 22.12 kVAr) at the end. The lowest voltage is raised to 0.9821 p.u. on the 47th bus, which is the lowest conceivable. As a result, the AVD and VSI in this investigation were found to be 0.00112 and 0.9301, respectively. The grid dependency for real power is 10.27%, which is lower than the grid dependency for uncompensated power, according to the optimal allocation of SPV systems with a total installed capacity of 2442.55 kW. The results of AOA for various SPV_IC levels are shown in Table 8. For comparison, the results of different HSAs for 100% SPV_IC levels are also shown.

Table 8:

Results of 85-bus system with agriculture loads

SPVIC (%)	Algorithm	Locations			SPV Sizes (kW)			Power Factors			Ploss (kW)	Qloss (kW)	Vmin (p.u.)	AVD	VSI	Griddep (%)
0	-	-			-			-			264.01	166.09	0.8825 (54)	0.00976	0.6066	100%
50	AOA	40	62	49	451.79	398.76	370.72	0.864	0.856	0.860	90.90	57.56	0.9394 (76)	0.00485	0.7788	54.17
60	AOA	49	10	60	639.02	146.85	679.66	0.863	0.866	0.861	72.42	45.06	0.9513 (76)	0.00413	0.8189	45.30
70	AOA	34	66	76	805.13	757.74	146.91	0.868	0.857	0.967	60.49	37.24	0.9598 (84)	0.00314	0.8488	36.52
80	AOA	34	79	25	753.81	611.03	589.21	0.875	0.885	0.893	52.47	31.86	0.9655 (84)	0.00271	0.8688	27.87
90	AOA	10	72	48	459.81	714.00	1024.49	0.903	0.855	0.864	44.79	25.20	0.9767 (84)	0.00151	0.9100	19.00
100	AOA	51	11	66	592.83	1070.65	779.07	0.850	0.858	0.858	39.70	22.12	0.9821 (47)	0.00112	0.9301	10.27
100	PSO	85	67	31	208.15	825.27	1409.13	0.932	0.859	0.886	45.782	24.947	0.9828 (84)	0.00093	0.9329	10.47
100	FPA	9	59	44	899.96	894.96	647.63	0.864	0.851	0.876	43.082	23.223	0.9783 (76)	0.00157	0.9161	10.42
100	TLBO	64	31	28	1126.37	1197.39	118.79	0.860	0.850	0.909	42.578	22.691	0.9815 (84)	0.00087	0.9279	10.28
100	CSA	28	66	52	968.04	869.85	604.67	0.867	0.855	0.886	42.537	23.292	0.9803 (84)	0.00097	0.9236	10.34
100	ALO	67	9	48	831.48	1146.58	464.49	0.895	0.856	0.894	42.432	23.890	0.9742 (47)	0.00171	0.9008	10.50
100	GHO	35	72	8	545.89	828.02	1068.65	0.919	0.861	0.854	41.435	23.101	0.9763 (54)	0.00162	0.9084	10.46

Scenario 3: Composite loads

This scenario, which is explained in more detail in Section 1.1.2, assumes that each bus carries a load that is 50% residential, 30% commercial and 20% industrial, with the remaining 20% being industrial. Both the loss and the AVD are optimized simultaneously in this circumstance (i.e. OF₂). When voltage-dependent load modelling is applied to compute the load flow, the overall agriculture loading effect equals (2395.528 kW + j 1926.514 kVAr) and the total loss equals (199.668 kW + j 125.845 kVAr). The bus with the lowest voltage, 0.8960 p.u., is on the 54th bus of the entire feeder. As a result, the AVD and VSI are found to be 0.00864 and 0.6446, respectively. For your convenience, the AOA findings for various SPVIC levels are shown in Table 9. In addition, the outcomes of other HSAs are compared to the outcomes of the HSA with 100% SPV_IC.

Table 9:

Results of 85-bus system with composite loads

SPVIC (%)	Algorithm	Locations			SPV Sizes (kW)			Power Factors			Ploss (kW)	Qloss (kW)	Vmin (p.u.)	AVD	VSI	Griddep (%)
0	-	-			-			-			199.68	125.85	0.8960 (54)	0.00864	0.6445	100%
50	AOA	36	69	34	156.90	597.76	443.11	0.871	0.864	0.878	76.41	48.66	0.9475 (54)	0.00451	0.8059	53.90
60	AOA	69	77	35	529.70	151.73	755.88	0.867	0.923	0.862	62.25	39.32	0.9555 (84)	0.00362	0.8335	45.46
70	AOA	64	29	69	565.71	911.13	200.02	0.884	0.859	0.882	54.56	32.98	0.9539 (54)	0.00340	0.8279	37.07
80	AOA	72	28	45	785.88	480.04	650.50	0.895	0.851	0.852	47.34	27.74	0.9677 (84)	0.00229	0.8768	28.59
90	AOA	67	31	9	506.44	864.74	784.80	0.859	0.886	0.868	38.94	22.62	0.9708 (54)	0.00218	0.8881	20.24
100	AOA	28	63	48	607.29	1159.40	628.85	0.851	0.867	0.852	34.06	18.94	0.9812 (84)	0.00096	0.9267	11.60
100	GOA	72	29	41	1154.87	733.72	506.94	0.853	0.850	0.860	44.32	22.75	0.9819 (84)	0.00099	0.9295	11.57
100	CSA	73	57	30	113.87	1173.48	1108.17	0.940	0.868	0.869	42.83	23.03	0.9747 (76)	0.00160	0.9027	11.80
100	ALO	8	64	53	755.20	911.50	728.83	0.905	0.862	0.864	41.10	22.73	0.9796 (84)	0.00125	0.9209	11.76
100	TLBO	72	32	57	885.01	1199.13	311.40	0.865	0.860	0.898	39.42	21.51	0.9806 (84)	0.00096	0.9247	11.66
100	FPA	60	31	7	798.09	1041.96	555.48	0.861	0.863	0.865	39.28	22.35	0.9735 (76)	0.00198	0.8980	11.72
100	PSO	72	8	33	775.96	1001.46	618.11	0.861	0.854	0.885	35.95	19.93	0.9728 (54)	0.00164	0.8956	11.74

The ideal SPV systems for a 100% SPVic level are 607.29 kW, 1159.4 kW, 628.85 kW at buses 28, 63 and 48, respectively. The overall losses were reduced to (34.06 kW + 18.94 kVAr) as a result. The lowest voltage is raised to 0.9812 p.u., which is the greatest voltage, on the 84th bus. As a result, the AVD and VSI in this investigation were found to be 0.00096 and 0.9267, respectively. The grid dependency of the system for real power is 11.60%, which is lower than the grid dependency of the uncompensated system of 11.60%, according to the optimal allocation of SPV systems with a total installed capacity of 2395.528 kW. Table 8 displays the AOA findings for various SPV_IC levels as well as various AOA levels. Other results of HSAs for 100% SPV_IC levels are also offered for comparison.

4.2 Discussion considering environmental aspects

This section details the reduced GHG emissions achieved through optimal SPV-system allocation for all situations in each RDS operation. In this analysis, the grid power is assumed for conventional thermal power plants and correspondingly different pollutants are computed. Based on the grid power delivered before and after the allocation of SPV systems, the GHG emissions are computed and shown in Table 10. The total power demand of the feeder on the grid, including losses, is 672.56 kW, resulting in a total GHG output of 1377.18 lb. The overall power consumption of a 22-bus feeder on the grid, including losses, is 672.56 kW. The grid dependency, including losses, is 5.79%, or 38.91 kW, resulting in a total GHG emission of roughly 79.42 lb, or a 94.23% reduction in GHG emissions. In a 28-bus rural feeder, true power grid dependence is ~763.76 kW in both scenarios, although it is decreased to 18.04 and 18.17 kW in Scenarios 1 and 2, respectively. As a result of this reduction, the overall GHG emissions are reduced by 97.65% in Scenario 1 and 97.63% in Scenario 2, respectively. Similarly, the grid dependency for Scenarios 1, 2 and 3 in an 85-bus feeder is 288.61, 2706.56 and 2595.20 kW, respectively, in the first, second and third scenarios. The output is lowered to 99.30, 29.70 and 34.06 kW if an SPV is installed. As a result, the grid deficiency in Scenarios 1, 2 and 3 is roughly 3.44%, 1.47% and 1.31%, respectively. In total GHG emissions, Scenarios 1, 2 and 3 have identical percentages of 96.57%, 98.54% and 98.69%. As can be seen in the findings, grid dependency and, as a result, GHG emissions are greatly decreased with the appropriate deployment of SPV systems.

Table 10:

Results of GHG emission for 100% SPV penetration levels

RDS	Before SPV Systems					After SPV Sysems					GHG Reduction (%)	Griddep (%)
	Pgrid (kW)	NOx (kg/h)	SO2 (kg/h)	CO2 (kg/h)	Total GHGem (kg/h)	Pgrid (kW)	NOx (kg/h)	SO2 (kg/h)	CO2 (kg/h)	Total GHGem (kg/h)
22-bus	672.56	1.544	3.539	619.593	624.676	38.91	0.089	0.205	35.846	36.140	94.23	5.79
28-bus (Scenario—1)	763.76	1.753	4.019	703.611	709.383	18.04	0.041	0.095	16.619	16.756	97.65	2.36
28-bus (Scenario—2)	763.76	1.753	4.019	703.611	709.383	18.17	0.042	0.096	16.739	16.876	97.63	2.38
85-bus (Scenario—1)	2886.21	6.624	15.186	2658.909	2680.720	99.30	0.228	0.522	91.480	92.230	96.57	3.44
85-bus (Scenario—2)	2706.56	6.212	14.241	2493.408	2513.861	39.70	0.091	0.209	36.573	36.873	98.54	1.47
85-bus (Scenario—3)	2595.20	5.957	13.655	2390.818	2410.429	34.06	0.078	0.179	31.378	31.635	98.69	1.31

5 Conclusion

SPVs are ideally sized and installed at optimal locations, according to the conclusions of this article, reducing the reliance on the grid and, as a result, GHG emissions from conventional power generation by utilizing a novel and recently developed nature-inspired metaheuristic multi-objective AOA. The search space for SPV systems is predetermined using LSFs, and AOA is used to derive ideal locations and associated sizes towards the objective function under various planning and operational limitations, both equal and unequal in nature, in order to achieve the goal. Simulated studies on 22-bus agriculture, 28-bus rural and 85-bus urban feeders in India have demonstrated the technical benefits of using separate solar-powered feeders in grid operation, as well as the environmental benefits of reducing GHG emissions from conventional power generation by using solar-powered feeders. When compared to existing optimization algorithms such as PSO, FPA, TLBO, CSA, ALO and grasshopper optimization algorithm (GOA), the suggested AOA is evaluated in terms of its effectiveness in solving multi-constraint optimization issues. The AOA is outperformed in all circumstances in terms of reduced objective function and redundancy in convergence characteristics, as well as in terms of convergence characteristics, according to the comparison analysis.

Conflict of interest

None declared.