An improved meta-heuristic method for optimal optimization of electric parking lots in distribution network

Abstract

In this study, a stochastic multi-objective structure for optimization of the intelligent electric parking lots (EPLs) is implemented in the distribution network for minimizing the power losses annual costs, power purchased from the main grid, unsupplied energy of subscribers, cost of vehicles to the grid as well as minimizing the network voltage deviations considering battery degradation cost (BDC) and network load uncertainty (NLUn). In this research, the unscented transformation method (UTM) is used for NLUn modeling and this method is easily applicable and has a low computational cost. An improved meta-heuristic algorithm named improved fire hawks optimization (IFHO) is utilized for decision variables finding defined as the site and size of the EPLs in the distribution network. The conventional fire hawks optimization (FHO) algorithm is inspired by the fire hawks foraging behavior and in this research, the Taylor-based neighborhood technique (TBNT) is used to reduce the dependency and the possibility of becoming trapped in local optimal. To evaluate the proposed methodology, the simulations are implemented in three scenarios (1) EPLs optimization without BDC and NLUn based-UTM, (2) EPLs optimization with BDC and without NLUn, and (3) EPLs optimization with BDC and NLUn. The results of the third scenario considering BDC and NLUn showed that the EPLs optimization integrated with a multi-objective framework by finding the EPL's optimal size and capacity in the network via the IFHO has reduced the annual losses, voltage deviations, ENS cost, and substation cost by 21.06%, 12.15%, 70.82%, and 39.10%, respectively compared to the base distribution network. Additionally, the results demonstrated that incorporating the BDC and NLUn, the annual losses, voltage oscillations, ENS cost, grid cost, and EPLs have increased in comparison with the EPLs optimization without BDC and NLUn based-UTM. In addition, the TBNT based-IFHO superiority has been confirmed in different scenarios by achieving better values of the objectives and also obtaining the convergence process with lower convergence tolerance and higher convergence accuracy.

Introduction

Motivation and background

The transition towards sustainable energy systems is characterized by the increasing linking of plug-in electric vehicles (PEVs) into the transport infrastructure¹. This change is not only a step towards reducing carbon emissions but also creates new challenges and opportunities for electricity distribution networks². A critical component of this integration is the development and implementation of plug-in electric vehicle parking. PEVs play a considerable role to improve the effectiveness of electricity distribution networks by acting as manageable loads and potential storage sources³. Through smart charging strategies, EPLs can help balance demand and supply, reduce peak loads, and increase energy system stability⁴. Moreover, it is very important to incorporate network reliability as a measure of the unsupplied energy of subscribers in the field of EPL integration⁴. In addition, modeling the network load uncertainty is another critical aspect of optimizing the EPLs⁵. Uncertainty in electricity demand can significantly affect the network operation. Accurate modeling of these uncertainties is the key to developing robust charging programs that can adapt to different conditions and ensure the efficient operation of the electric grid.

Optimization and planning of electric plug-in EPLs in electricity distribution networks are complex processes that include multiple objectives and limitations. To deal with these challenges, meta-heuristic algorithms⁶ have emerged as a powerful tool to find the site and size of the installation and operation of EPLs in the networks, optimally^7,8. By exploring a vast search space for possible solutions, meta-heuristic algorithms can identify the optimum installation site and size of PEVs, increase network reliability, and accommodate inherent uncertainties in electrical distribution networks. Finally, integrating PEVs in electricity distribution networks facilitates daily life by achieving sustainable energy systems.

Related work and research gap

Numerous studies using various objective functions and solvers have been conducted on the topic of EPL optimization in networks. To determine the number, site, and ideal size of EPLs while considering the uncertainty of distributed generation, a fuzzy grasshopper optimization technique for EPLs is proposed in⁹. A strategy for utilizing EPLs to optimize revenue using their involvement in different energy and auxiliary services sectors is provided in¹⁰. A probabilistic approach is proposed in¹¹ for unit commitment in energy hubs with EPLs to lower operational costs while taking the response to demand into account. A novel accumulating agency for the energy system is introduced in¹² for maximizing the EPLs' profit-maximizing involvement in various market segments for energy. A mixed integer linear programming approach is provided in¹³ for microgrid management using EPLs to maximize microgrid revenues by optimizing the network. To decline the overall cost, which includes the power cost of the substation, the cost of electricity losses, the cost of charging and releasing EVs, and the cost of supplies, an arithmetic optimization approach is described in¹⁴ to calculate the optimal size and position of IELPs. In¹⁵, a linear mixed integer programming considering the uncertainties related to EV owners' activity is used to offer a probabilistic optimal position and capacity of EPLs with the goal of lowering distribution network losses and boosting dependability. A two-stage optimal planning optimization that takes into account both the highest level network energy uncertainty and EV activity uncertainties is the one proposed in¹⁵. Conditional value at risk is used in¹⁶ as part of an approach to risk assessment to measure the effects of high uncertainty on EPL optimization. This is a helpful instrument for decision-making when it comes to reducing the risks linked with variations in energy prices. In¹⁷, multi-objective optimization structure is presented for optimal planning of wind energy resources and EPLs in the power system incorporating the EVs load, and wind power uncertainties to maximize the voltage stability, and minimize the cost, and emissions using a point estimate method (PEM). To achieve the greatest revenues, the positioning of EPLs in a distribution network is discussed in¹⁸ taking into account the charge/discharge activity of EVs using an efficient probabilistic. To optimize the owners' overall earnings while meeting the greatest demand for parking plug-in electric vehicles (PEVs), an allocation framework for the best location, capacity, and charging optimization of EPLs is described in¹⁹. Utilizing the network lines outage data,²⁰ assesses the effects of EPLs on the network dependability by minimizing energy not served. The load probability model, the production of the EPLs, and an appropriate calculation of the annual hourly demand utilizing a cuckoo optimization algorithm (COA) and Monte Carlo simulation are taken into account in²¹ to accomplish the best possible positioning of EPLs. The allocation of EPLs is performed, and their installation position and size are discovered, by applying the Monte Carlo simulation and genetic algorithm (GA) in²². Without taking into account the battery charge pattern of the EVs,²³ performs the multi-objective positioning of EPLs to lower costs, increase voltage and dependability, and enhance reliability. In²⁴ a two-stage multi-objective optimization method is proposed for optimal planning of distributed generations in distribution networks to maximize the investment benefits and voltage stability and also minimize the losses, and the effect of energy storage is evaluated in the problem-solving. In²⁵, the optimization of the EPLs is looked at to produce optimal management for the greatest possible use of capabilities to determine the upper amount of EPLs that can be employed in the network and to expand the current size to satisfy the requirements for EVs. According to²⁶, battery degradation is taken into account while managing EVs at charging stations, taking into account the advantages for both users and operators. A strategy for the EV firm and charging EPLs to take part in the energy and reserve market is put forward in²⁷. To optimize the EPL's revenue,²⁸ evaluates the EV scheduling in the EPLs. The optimization of EV charging effect on the network is examined in²⁹, taking battery capacity and charging power into account. Establishing the EPLs in the electrical network is suggested in³⁰ to reduce installation costs while enhancing the voltage conditions and lowering the losses.

By evaluating the research done on the problem of optimizing EPLs in distribution networks, the research gaps are explained as follows:

• The optimization of EPLs in distribution networks has seen substantial research, focusing on various performance metrics. However, most studies overlook the cost of battery degradation in electric vehicles (EVs). Given the significant impact of battery degradation costs on overall operational costs, this omission presents a notable gap. The effect of ignoring battery degradation costs on the distribution network's performance remains underexplored, highlighting a critical area for future research to ensure more accurate and comprehensive optimization models for EPLs.

• Moreover, the uncertainty of load in distribution networks plays an essential role in the optimization of EPLs. Most existing models assume static load profiles, which simplifies the optimization problem but does not reflect the inherent unpredictability of real-world operations. This uncertainty can significantly impact the effectiveness and reliability of optimization solutions. Addressing this gap by incorporating load uncertainty into optimization models can lead to more robust and adaptable strategies for managing EPLs in electrical distribution networks.

• The Unscented Transformation (UT) method³¹ offers several advantages over conventional probabilistic and stochastic methods for uncertainty modeling. UT provides more accurate and computationally efficient estimates of the mean and variance of a distribution, especially for non-linear systems common in electrical networks. This accuracy and efficiency make UT a promising approach for improving the modeling of uncertainties in the optimization of EPL, potentially leading to more reliable and effective optimization strategies in comparison with those based on classical probabilistic or stochastic methods.

• Finally, the application of new meta-heuristic algorithms in optimizing EPLs in distribution networks is necessitated by the No Free Lunch (NFL) theorem³². This theorem suggests that no single optimization algorithm performs best for all possible problems. Therefore, exploring new or improved meta-heuristic algorithms tailored to the unique challenges of EPL optimization can offer new solutions. This approach can potentially overcome the limitations of existing algorithms and improve optimization outcomes, underscoring the importance of continuous algorithmic innovation in this field.

Paper's contributions

In terms of the described research gaps, the contributions of this study are presented in following.

• In this research, a stochastic and multi-objective structure for EPLs optimization in the network is proposed for minimizing the costs of annual losses, received power from the post, unsupplied energy of subscribers, cost of vehicles to the grid as well as minimizing the network voltage deviations considering vehicles battery degradation cost (BDC).

• A major challenge in developing precise EPLs optimization in the network lies in accounting for battery degradation during the optimization process because of the battery degradation is nonlinear and depends on multiple independent variables, some of which are difficult to compute within the same time domain as other application variables such as batter depth of discharge (DOD).

• An unscented transformation method (UTM) is applied to model the network load uncertainty (NLUn) for stochastic EPLs multi-objective optimization in the distribution network.

• A new improved fire hawks optimization (IFHO) is proposed to solve the EPLs optimization in the network and find the site and size of the EPLs in the network, optimally. In the improved algorithm, the Taylor-based neighborhood technique (TBNT)³³ is applied to reduce the dependency and the possibility of becoming trapped in local points of the conventional FHO³⁴.

• Performance of the TBNT-based- IFHO to solve the EPLs optimization is compared with conventional FHO, PSO, GWO, MRFO, and AEFA methods.

Paper's structure

The problem formulation, which covers the multi-objective function and the problem restrictions, is covered in the section that follows. Section “Problem formulation” presents an overview of the TBNT-based IFHO and its use. In section “Proposed optimization method”, the UTM based-NLUn is developed. The results of the used method are shown in sections “UTM based-NLUn modelling”, and “Simulation results and discussions” summarized and discussed the findings.

Problem formulation

This section develops the suggested problem formulation for the EPLs optimization problem, together with the objective function and restrictions.

Objective function

The objective function (OF) of optimization the EPLs in distribution network is defined as for minimizing the costs of annual losses, power obtained from the substation, unsupplied energy of subscribers, the cost of vehicles to the grid and to minimize the total voltage deviations of the grid as follows based on the multi-objective approach of weighted coefficients. According to the multi-objective method of weighted coefficients, the OF for optimizing the EPLs in the distribution network is presented as follows:

$${min \,F}_{T}=\left({\widetilde{\Psi }}^{1}\times \frac{{C}_{loss}}{{C}_{loss,max}}\right)+\left({\widetilde{\Psi }}^{2}\times \frac{{V}_{d}}{{V}_{d,max}}\right)+\left({\widetilde{\Psi }}^{3}\times \frac{{C}_{subp}}{{C}_{subp,max}}\right)+\left({\widetilde{\Psi }}^{4}\times \frac{{C}_{ens}}{{C}_{ens,amx}}\right)+({\widetilde{\Psi }}^{5}\times \frac{{C}_{IEPL}}{{C}_{IEPL,amx}})$$

(1)

where, ${C}_{loss}$, ${V}_{d}$, ${C}_{subp}$, ${C}_{ens}$, and ${C}_{IEPL}$ stand for the annualized losses cost, total voltage deviation, annual purchased grid power cost, annual energy not-supplied cost, and annual EPLs cost, respectively. Also, ${C}_{loss,max}$, ${V}_{d,max}$, ${C}_{subp,max}$, ${C}_{ens,max}$, and ${C}_{v2g,max}$ are upper limit of ${C}_{loss}$, ${V}_{d}$, ${C}_{subp}$, ${C}_{ens}$, and ${C}_{IEPL}$, respectively. Moreover, ${\widetilde{\Psi }}^{1}$, ${\widetilde{\Psi }}^{2}$, ${\widetilde{\Psi }}^{3}$, ${\widetilde{\Psi }}^{4}$, and ${\widetilde{\Psi }}^{5}$ are weighted coefficients of different goals of ${C}_{loss}$, ${V}_{d}$, ${C}_{subp}$, ${C}_{ens}$, and ${C}_{IEPL}$, respectively.

Cost of annual losses

The network power losses may be declined by the EPLs optimization in the network. The annual losses cost is computed as follows^7,8,9,10:

$${C}_{loss}=365{\times \gamma }_{l}\times \sum_{m=1}^{M}\sum_{n=1}^{N}{[\alpha }_{mn}\left({P}_{m}{P}_{n}+{Q}_{m}{Q}_{n}\right)+{\beta }_{mn}\left({Q}_{m}{P}_{n}-{P}_{m}{Q}_{n}\right)]$$

(2)

$${\alpha }_{mn}=\frac{{r}_{mn}}{{V}_{m}{V}_{n}}\text{cos}({\delta }_{m}-{\delta }_{n})$$

(3)

$${\beta }_{mn}=\frac{{r}_{mn}}{{V}_{m}{V}_{n}}\text{sin}({\delta }_{m}-{\delta }_{n})$$

(4)

where, ${Z}_{mn}={r}_{mn}+{jx}_{mn}$ refers to the line impedance connecting bus m to bus n, P and Q stand for the reactive as well as the real power at each bus, respectively, and m and n denote the transmitting and getting bus indexes, respectively.

Voltage deviation

One of the distribution network operator's main responsibilities, which is also a consumer issue, is to improve voltage conditions. Thus, when adding EPLs, it is imperative to investigate the voltage oscillation as computed by^35,36

$${V}_{d}=\sqrt{\frac{1}{M}\times {\sum }_{m=1}^{M}({v}_{i}-{v}_{p}{)}^{2}}$$

(5)

$${v}_{p}=\frac{1}{M}\times {\sum }_{m=1}^{M}{v}_{i}$$

(6)

where, v_i denotes the ith bus voltage, v_p is the mean bus voltage, and M refers to the buses number.

Cost of purchased power from the substation

One of the advantages of optimizing the EPLs in the network is freeing the network capacity and decreasing the power purchased from the substation. The value of electricity received from the substation multiplied by the electricity price per kW is defined as follows:

$${C}_{subp}=\sum_{n=1}^{T}({P}_{sub}\times {c}_{sub})$$

(7)

where, ${P}_{sub}$ denotes the active power obtained from the substation, and ${c}_{sub}$ clears the electricity price for each kW.

Cost of energy not-supplied

Under various operating situations, the distribution network's reliability is a critical issue that needs to be improved. Consequently, throughout the study, the energy not supplied (ENS) as a reliability index is chosen and taken into account. The following formula is applied to compute the annual ENS cost^7,37:

$${C}_{ens}=\sum_{b=1}^{l}(\sum_{n=1}^{T}({\sum }_{h}(\frac{\varpi (h)}{T}\times {\partial }_{f}\times {d}_{out})\times {\tau }_{ns}))$$

(8)

where, l clears network lines number, $\varpi \left(h\right)$ denotes the mean load at time h, ${\partial }_{f}$ refers to the line failure value at h, ${d}_{out}$ is duration of outage at h, and ${\tau }_{ns}$ refers to the cost of per kWh not-met energy ($/kWh).

Cost of EPLs

The three primary components that make up the cost of V2G are capital, wear, and bought energy. Bought energy and wear charges are additional expenses required for V2G, not for driving a vehicle. On the other hand, the installed technology needed for V2G represents the capital cost. The following is an expression of the three parts of V2G cost^38,39:

$${C}_{EPL}={C}_{cap}\left(i\right)+{C}_{driving}\left(i\right)+{C}_{v2g}\left(i\right)+{C}_{deg}\left(i\right)$$

(9)

$${C}_{cap}\left(i\right)={c}_{ac}\times PC(i)$$

(10)

$${C}_{driving}\left(i\right)=\sum_{j=1}^{tn}(\left(\frac{{p}_{off}}{{\mu }_{conv}}\right){P}_{IEPLch}\left(i,j\right)\times t(j))$$

(11)

$${C}_{v2g}\left(i\right)=(\frac{{p}_{off}}{{\mu }_{conv}}+{c}_{d})\times {P}_{IEPL}\left(i\right)\times {t}_{disp}$$

(12)

where, ${C}_{cap}\left(i\right)$ denotes the capital cost of ith EPL, ${c}_{ac}$ refers to the each EV annual cost, $PC(i)$ is the size of ith EPL, ${C}_{driving}\left(i\right)$ clear the cost of purchased power for charging the EV driving, ${p}_{off}$ denotes the electricity cost at off-peak hours, ${c}_{d}$ refers to wearing cost of devices, ${P}_{EPLch}\left(i,j\right)$ denotes the needed power for charging the EVs (0%-100%) during of $t(j)$, ${\mu }_{conv}$ represents the inverter efficiency, and ${C}_{v2g}\left(i\right)$ clears the purchased energy cost for charging the EVs for V2G power. In the present study, the variability in EPLs charging demands are accommodated using the main grid power, and also with handling the optimization framework⁵.

Battery degradation can be categorized into two main types: Calendar Life (CaL) and Cycle Life (CiL)^40,41. The CaL refers to the natural decline in battery capacity over time, independent of the number of charge and discharge cycles. It is influenced by environmental conditions at the battery's installation site and can be seen as a non-operational factor. The CiL on the other hand, represents the maximum number of full charge and discharge cycles a battery can undergo. This type of degradation is primarily driven by operational factors, such as the frequency and depth of charging and discharging. In summary, the cost of battery degradation directly affects the cost of charging and discharging because it influences how quickly a battery degrades and how often it needs to be replaced, leading to increased operational and replacement costs over the battery's lifetime. As the battery wears out, its efficiency decreases, leading to less energy being stored or delivered per cycle. This inefficiency requires more frequent charging, resulting in higher energy costs. Additionally, as the battery ages, maintenance and management costs can also rise, further increasing the overall cost of operation.

Although CaL and CiL are interrelated to some extent, our study focuses mainly on CiL, as it aims to optimize the Battery Energy Storage System (BESS) operating plan for the Electric Power Line (EPL) over a specific time horizon by addressing the Battery Degradation Cost (BDC). Since CaL is a non-operational factor, the study does not account for time-varying financial elements such as interest and inflation^40,41.

As the BESS operates, the CiL of an EPL gradually diminishes. However, our research primarily considers the impact of Depth of Discharge (DOD), which is crucial for optimizing the usage of EPLs within the network. The State of Charge (SOC) also influences battery degradation, particularly at very high or low levels, but this effect can be managed through optimization constraints. Operating at a high DOD significantly reduces the maximum number of full cycles, which is a key aspect of CiL. The BDC is calculated based on these considerations⁴⁰.

$$BDC=\frac{{C}_{I}\times DOD\times {E}_{Stor}}{{N}_{cycle}}$$

(13)

whereas ${E}_{Stor}$ displays the battery's useful power and ${C}_{I}$ indicates the battery investment. The battery DOD is typically determined by its state of charge (SOC) and is computed as the proportion of the battery's charge to its overall charge size, where n is the greatest charge that the battery can hold. The DOD is calculated by.

$$DOD=\frac{{E}_{Stor,max}-{E}_{i}}{{E}_{Stor,max}}$$

(14)

where, ${E}_{i}$ indicates the energy storage inside the battery per hour and ${E}_{Stor,max}$ denotes the maximum energy storage within the battery. The following is another way for obtaining the battery CiL:

$${N}_{cycle}=\alpha \times {DOD}^{\beta }$$

(15)

where, α and β represent certain battery characteristics that were obtained through experimentation based on each battery's manufacturer's assessment. The corresponding values of α and β for a lithium-ion battery in this study are 1331 and -1.825⁴⁰, respectively⁴⁰.

Constraints

Power flow constraint

The distribution network load flow limitations are expressed as follows^{7,8,9,10,36,37}:

$${P}_{sub}+\sum_{i=1}^{{N}_{IEPL}}{P}_{EPL}\left(i\right)=\sum_{b=1}^{{n}_{b}}{P}_{loss}\left(b\right)+\sum_{m=1}^{{n}_{d}}{P}_{dmd}\left(m\right)$$

(16)

$${Q}_{sub}=\sum_{b=1}^{{n}_{b}}{Q}_{loss}\left(b\right)+\sum_{m=1}^{{n}_{d}}{Q}_{dmd}\left(m\right)$$

(17)

where, ${P}_{sub}$ and ${Q}_{sub}$ refer to the slack bus active and reactive powers,${n}_{b}$ and ${n}_{d}$ denote the demand buses and lines number, ${P}_{loss}\left(b\right)$ and ${Q}_{loss}\left(b\right)$ are the branch b active and reactive losses, ${P}_{dmd}\left(m\right)$ and ${Q}_{dmd}\left(m\right)$ clear the bus m active and reactive loads.

Bus voltage constraint

Each bus's voltage of operation needs to meet each bus's allowable ranges. The branch admission is denoted by ${Y}_{ij}$. The bus voltage limit is expressed by

$${V}_{min}\le {V}_{m}\le {V}_{max} , m=\text{1,2},\dots ,M$$

(18)

where, ${V}_{m}$ denotes the bus m present voltage and ${V}_{min}$ and ${V}_{max}$ are equal to 0.95 and 1.05 respectively.

Line flow constraint

The line flow must be lower than the greatest size permitted by the line as follows:

$${S}_{LB}\le {S}_{LB}^{max}$$

(19)

where, ${S}_{LB}^{max}$ denotes the maximum allowable capacity of the network line.

Substation power constraint

The imported energy cost into the network from the substation is expected to rise by 10% for each period in which the power purchased surpasses the maximum allowed amount (${P}_{sub,max}$). In this study, it is assumed that the upper grid power limit is 30% of the network's total maximum load. The following is the cost restriction:

$${c}_{sub}={c}_{1 }\forall \,{P}_{sub}\le {P}_{sub,max}$$

(20)

$${c}_{sub}={c}_{2 }\forall \,{P}_{sub}>{P}_{sub,max}$$

(21)

where, ${c}_{2}=1.1\times {c}_{1}.$

EPL batteries storage capacity constraint

The batteries manufactured by EPL should meet the following lower and upper criteria for SOC:

$${SOC}_{min}\le {SOC}_{i}\le {SOC}_{max}$$

(22)

where, ${SOC}_{min}$ and ${SOC}_{max}$ are lower and upper level of the EPL's SOC.

EPL's rates of charge and discharge constraint

$${P}_{ch}\le {R}_{ch}$$

(23)

$${P}_{dch}\le {R}_{dch}$$

(24)

EPL's capacity constraint

The network's installed EPL size shouldn't be greater than its upper size, which is defined by

$${P}_{EPL}\le {P}_{IEPL}^{max}$$

(25)

where, ${P}_{IEPL}^{max}$ is upper limit of the EPL.

Proposed optimization method

This section presents an illustration of the FHO's inspiration notion and also the mathematical model of the suggested meta-heuristic method. As part of long-standing cultural and ethnic traditions, Aboriginal Australians use fire as a useful tool for management and preserving the equilibrium of the surrounding ecology and territory. Additionally, brown hawks are to blame for the nationwide spread of fires. These birds, called fire hawks, intentionally spread fire by tossing flaming twigs into other unburned areas to ignite little fires that frighten individuals. They do this by clutching the sticks in their mouths, claws snakes, rats, and other creatures, making them flee in the stress and hurry of these creatures thereby rendering it much simpler for birds of prey to capture.

Fire Hawk optimizer (FHO)

The FHO meta-heuristic approach takes into account the steps of beginning and expanding fires by collecting prey, and imitating the behavior of fire hawks foraging. First, the location vectors related to the hawks and prey are identified, leading to several answer candidates (X). The beginning coordinates of these vectors in the area related to the space of search are determined via an arbitrary starting procedure³⁴.

$$X=\left[\begin{array}{c}{X}_{1}\\ {X}_{2}\\ \vdots \\ {X}_{i}\\ \vdots \\ {X}_{N}\end{array}\right]= \left[\begin{array}{cc}\begin{array}{cc}{x}_{\text{1,1}}& {X}_{\text{1,2}}\\ {x}_{\text{2,1}}& {X}_{\text{2,2}}\end{array}& \begin{array}{ccc}\cdots & \cdots & {x}_{1,\text{j }}{ x}_{1,\text{d}-1} { x}_{1,\text{d}}\\ \cdots & \cdots & {x}_{2,\text{j }}\dots { x}_{2,\text{d}}\end{array}\\ \begin{array}{cc}\vdots & \vdots \\ \vdots & \vdots \\ {x}_{\text{N},1}& {X}_{\text{N},2}\end{array}& \begin{array}{ccc}\vdots & \vdots & \vdots \vdots \vdots \\ \vdots & \vdots & \vdots \vdots \vdots \\ \dots & \dots & {x}_{\text{N},\text{j}} { x}_{\text{N},\text{d}-1} { x}_{\text{N},\text{d}}\end{array}\end{array}\right], \left\{\begin{array}{c}i=\text{1,2},\dots ,N\\ j=\text{1,2},\dots ,d\end{array}\right.$$

(26)

$${x}_{i}^{j}\left(0\right)={x}_{i,min}^{j}+rand.\left({x}_{i,max}^{j}-{x}_{i,min}^{j}\right), \left\{\begin{array}{c}i=\text{1,2},\dots ,N\\ j=\text{1,2},\dots ,d\end{array}\right.$$

(27)

In this scenario, X_i denotes the ith answer candidate within the space of search; d signifies the issue's dimension; N is the answer individuals overall number within the search space; ${x}_{i}^{j}$ signifies the ith answer candidate's jth selection variable; ${x}_{i}^{j}(0)$ denotes the first two answer candidates' initial location; ${x}_{i,min}^{j}$ and ${x}_{i,max }^{j}$ are indicate the lowest and highest restrictions of the jth variable of the ith answer applicant; rand refers to a number that is evenly distributed within the [0,1] range, randomly.

The selected optimization problem is incorporated in the OF inquiry for the response alternatives to determine the locations of the Hawks in the space of search. Whereas the remaining answer competitors are the food, some candidates with higher OF quantities are presented as Fire Hawks. The designated Fire Hawks are used to disperse fires surrounding the target in the search area to facilitate hunting. Moreover, the initial fire that the Fire Hawks utilize to disseminate fires everywhere the area under search is believed to be the best option globally. The following is how these behavioral features are expressed³⁴:

$$PR=\left[\begin{array}{c}{PR}_{1}\\ {PR}_{2}\\ \vdots \\ {PR}_{l}\\ \vdots \\ {PR}_{n}\end{array}\right], k=\text{1,2},\dots ,m$$

(28)

$$FH=\left[\begin{array}{c}{FH}_{1}\\ {FH}_{2}\\ \vdots \\ {FH}_{l}\\ \vdots \\ {FH}_{n}\end{array}\right], l=\text{1,2},\dots ,n$$

(29)

where, FH_l represents the lth fire hawk taking into account the overall amount of n hawks in the space of search, and PR_k is the kth prey in the space of search concerning the total of m preys. The overall distance between the Fire Hawks and the prey is computed in the algorithm next step. An example of this viewpoint is shown in Fig. 1, where ${D}_{k}^{l}$ is calculated using the following formula³⁴:

$${D}_{k}^{l}=\sqrt{{({x}_{2}-{x}_{1})}^{2}+{({y}_{2}-{y}_{1})}^{2}}, \left\{\begin{array}{c}l=\text{1,2},\dots ,n\\ k=\text{1,2},\dots ,m\end{array}\right.$$

(30)

where m and n are the prey and fire hawks number in the area of search, respectively, and (x₁, y₁) and (x₂, y₂) are the prey's and fire hawks' positions within the searching area. Additionally,${ D}_{k}^{l}$ is the overall distance within the lth fire hawk and the kth prey.

Fig. 1

Diagrammatical depiction of how far the Fire Hawks are from their target overall³⁴.

The diagrammatic representation of identifying the Fire Hawks' territory in the search space is illustrated in Fig. 2a. The Fire Hawks gather flaming twigs from the main fire in the following algorithmic step to start a fire in the designated location. At this step, the prey is forced to quickly leave as each creature takes up a flaming stick and places it in the appropriate spot. The equation that follows³⁴ illustrates how these two behaviors can be used as position-updating methods in the principal search cycle of FHO since certain birds are anxious to employ the flaming sticks coming from other Fire Hawks' domains in the meantime:

Fig. 2

Diagrammatic representation of the steps involved in: (a) identifying the Fire Hawks' territory in the search space; and (b) updating their location within the area of search³⁴.

$${FH}_{l}^{new}={FH}_{l}+\left({r}_{1}\times GB-{r}_{2}\times {FH}_{Near}\right), l=\text{1,2},\dots ,n$$

(31)

In this case, r₁ and r₂ refer to the evenly dispersed with probabilities in a spectrum of (0, 1) to find the Fire Hawks motions in the major fire direction and the boundaries of the remaining Fire Hawks (Fig. 2b). GB is the universal optimum answer in the search area thought to be the main fire. ${FH}_{Near}$ denotes one of the different Fire Hawks in the space of search, and The lth Fire Hawk's (FH_l) fresh location vector is denoted by ${F}_{new}^{l}$.

The prey motion in the range for every Fire Hawk is considered a crucial component of animal behavior in the method's subsequent step, which updates the location of the animal. Whenever a Fire Hawk drops an ignited stick, its victim will either decide to run away, hide, or accidentally walk in the direction of the Fire Hawk. When changing the position, these movements can be taken into consideration using the equation that follows³⁴:

$${PR}_{q}^{new}={PR}_{q}+\left({r}_{3}\times {FH}_{l}-{r}_{4}\times {SP}_{l}\right), \left\{\begin{array}{c}l=\text{1,2},\dots ,n\\ q=\text{1,2},\dots ,r\end{array}\right.$$

(32)

where, GB represents the globally optimal answer in the search for space regarded as the primary fire; SP_l is the secure location beneath the lth Fire Hawk region; and r₃ and r₄ have equal distributions with random values in an interval of (0, 1) to calculate the motions of prey into the Fire Hawks and the position of the prey. In this scenario, ${PR}_{q}^{new}$ is the newly position vector of the qth prey (${PR}_{q}$) enclosed by the lth Fire Hawk (${FH}_{l}$).

Additionally, the prey might migrate towards the area of the other Fire Hawks, but it's also possible that they might approach the Fire Hawks in close captures may attempt to escape from their stuck position by trying to reach a more secure area outside of the Fire Hawk's realm. The Eq. (33)³⁴ can be used to take these actions into account while updating the location:

$${PR}_{q}^{new}={PR}_{q}+\left({r}_{5}\times {FH}_{Alter}-{r}_{6}\times SP\right), \left\{\begin{array}{c}l=\text{1,2},\dots ,n\\ q=\text{1,2},\dots ,r\end{array}\right.$$

(33)

whereas r₅ and r₆ are evenly dispersed randomly generated numbers in (0, 1) to calculate the motions of prey in the other Fire Hawks direction and a secure location beyond the boundaries, ${FH}_{Alter}$ is a Fire Hawk that is among the others in the search area. The mathematical formulation of SP_l and SP appears as following³⁴, founded on the observation that a secure area within nature is the place where most animals live congregate to stay secure and sound throughout a risk:

$${SP}_{l}=\frac{\sum_{q=1}^{r}{PR}_{q}}{r}, \left\{\begin{array}{c}q=\text{1,2},\dots ,r\\ l=\text{1,2},\dots ,n\end{array}\right.$$

(34)

In the area of search, the kth prey is denoted by ${PR}_{q}$.

The theoretical framework of this technique takes into account the fundamental characteristics of FHO, such as the requirement for termination and border infractions by solution candidates. To this end, the FHO implements a mathematical flag that determines the limit of control for the decision variables that are violated, and it can also use an established amount of OF assessments or iterations as a termination criterion.

Improved Fire Hawk optimizer (IFHO)

One of the primary limitations of the FHO such as the other metaheuristic algorithms is its tendency to get trapped in local optima and its limited ability to converge to a global solution in conditions of the high complexity of the problem. To address these challenges, integrating statistical methods into the FHO—resulting in an Improved FHO (IFHO)—can significantly enhance its optimization performance. This study introduces the Taylor-Based Neighborhood Technique (TBNT)³³ as a means to boost the effectiveness of FHO optimization. TBNT not only accelerates the FHO's early convergence but also expands the search space, thereby increasing the likelihood of discovering optimal solutions and improving the overall robustness of the algorithm.

The TBNT is an advanced generalization of the basic neighborhood method, extending it to the nth order. TBNT is rooted in the principles of Taylor expansion, which allows for a more in-depth exploration of the population within the search space compared to standard search methods. This approach broadens the learning scope of individuals, replacing the conventional location update based on globally optimal individuals with an optimal neighbor update technique. As a result, the Improved FHO (IFHO) benefits from enhanced local exploitation capabilities.

A key limitation of the original FHO is its lack of an efficient mechanism for thorough exploration of the search space during each iteration. TBNT addresses this by introducing Taylor-based neighborhoods that enable a more effective, comprehensive, and forward-looking assessment of the population. While meta-heuristic algorithms often rely on optimal individuals within the population, this reliance can increase the risk of the algorithm becoming trapped in local optima. TBNT mitigates this risk by reducing the algorithm's tendency to fixate on these points and decreasing dependency on them. In highly complex and non-convex optimization problems, the Taylor-based approach of TBNT enables the algorithm to converge more rapidly to feasible solutions, offering a robust and efficient strategy for navigating challenging search landscapes. This makes TBNT a valuable enhancement for IFHO, improving its performance in a wide range of optimization scenarios.

The TBNT enhances the characteristics and importance of the representative member and can help the representative member explore potential solutions nearby to expand the learning of the individual and improve the local exploitation situation. Following the application of the TBNT, the FHO location updating formulas are outlined below:

$${PR}_{q}^{new}=1+\alpha \left(\sigma \times R\left(\text{0,1}\right)\right)+\frac{\alpha \left(\alpha -1\right)}{2!}{(\sigma \times R\left(0.1\right))}^{2}+\dots +\frac{\alpha \left(\alpha -1\right)\dots \left(\alpha -n+1\right)}{n!}{(\sigma \times R\left(0.1\right))}^{n}\times {PR}_{q}+\left({r}_{5}\times {FH}_{Alter}-{r}_{6}\times SP\right), \left\{\begin{array}{c}l=\text{1,2},\dots ,n\\ q=\text{1,2},\dots ,r\end{array}\right.$$

(35)

It is evident from Eq. (35) that exceptional people are directly affected by the TBNT. In the interval [0,1], R(0,1) is an arbitrary number with a uniform distribution. The adjusting parameters σ and α are represented. The fundamental optimal neighborhood approach is realized when α = 0.6 is selected as the suggested efficient derivation ordering and the Taylor expansion, respectively. The neighborhood factor, represented by σ, indicates the neighborhood's size.

The integration of TBNT into conventional FHO significantly improves the improved algorithm (IFHO)'s ability to explore and exploit complex search spaces, making it a powerful tool for optimizing large-scale distribution networks. TBNT addresses the premature convergence by leveraging Taylor expansion principles to generalize the neighborhood search method to higher orders. This enables the algorithm to consider a wider range of potential solutions around each candidate, effectively broadening the search space. Unlike fixed neighborhood strategies, TBNT dynamically adjusts the neighborhood size based on the Taylor expansion. This flexibility allows the algorithm to adapt its exploration strategy according to the complexity of the problem, facilitating the discovery of global optima even in challenging search landscapes. The TBNT enhances the exploitation by allowing the representative member of the population to perform a more thorough local search around promising solutions. The nth-order Taylor expansion enables the algorithm to fine-tune the location of the candidate solutions more precisely, improving the convergence speed and the accuracy of the solution. As distribution networks grow in size and complexity, the search space becomes exponentially larger, making traditional optimization methods less effective. The TBNT’s ability to manage larger neighborhoods and adapt to the complexity of the problem allows IFHO to scale more effectively, maintaining performance even as the problem size increases. The enhanced exploration prevents the algorithm from becoming trapped in local optima, while the improved exploitation ensures that high-quality solutions are found more efficiently. This dual enhancement is particularly valuable in scenarios where the optimization problem is highly non-linear, multi-modal, and requires a balance between accuracy and computational efficiency.

Algorithm 1 presents the IFHO pseudo-code, and Fig. 3 shows its flowchart. Additionally, Fig. 4 displays a flowchart of the IFHO implementation to tackle the network's EPL optimization problem.

Fig. 3

Flowchart of the IFHO.

Fig. 4

Flowchart of the IFHO process to solve the EPLs optimization in the network.

Algorithm 1: Improved Fire Hawk Optimizer (IFHO).

UTM based-NLUn modeling

The unscented transformation method (UTM) for network load uncertainty (NLUn) modeling is used in this work. Because of its short computation time, UTM has been utilized extensively for uncertainty modeling in previous research³¹. For reducing the model complexity, no assumptions are also made. The ability of UTM to handle nonlinear transformations³¹ and provide highly appropriate estimates of probability distribution functions is well recognized. The input vector dimension for uncertainties (U) is indicated by the symbol n in this approach. n is fixed at 1 in this investigation. Furthermore, 2n + 1 possibilities are generated. Therefore, no scenario reduction approach is applied for this tiny number of situations, resulting in a large reduction in computation time.

y = f(x), where x ∈ Rⁿ refers to the uncertain source vector and y ∈ R^r denotes the uncertain outcome vector with r components, representing the issue with its uncertain, and stochastic characteristics. x's mean and covariance are represented by μ_x and σ_x. The symmetric and asymmetric components of x are applied to find the variance and covariance of uncertain parameter, respectively. The goal of applying the UTM is to find the results' mean and covariance values (μ_y and σ_y)³¹. The procedures below provide an explanation of this:

Step A) Select two samples 2n + 1 of the uncertain input parameter.

$$x_{0} = \mu_{x}$$

(36)

$$x_{s} = \mu_{x} + \sqrt {\frac{n}{{1 - W^{0} }}\sigma_{x} } \,\,\,\,\,\,\,\forall s = 1,2,...,n$$

(37)

$$x_{s} = \mu_{x} - \sqrt {\frac{n}{{1 - W^{0} }}\sigma_{x} } \,\,\,\,\,\,\,\forall s = 1,2,...,n$$

(38)

where, W⁰ denotes the mean quantity μ_x weight.

Step B) Examine each sample point's weighted factor individually:

$$W^{0} = W^{0}$$

(39)

$$W_{s} = \frac{{1 - W^{0} }}{2n}\,\,\,\,\,\,\,\forall s = 1,2,...,n$$

(40)

$$W_{s + n} = \frac{{1 - W^{0} }}{2n}\,\,\,\,\,\,\,\forall s + n = n + 1,n + 2,...,2n$$

(41)

$$\sum\limits_{s = 1}^{n} {W_{s} } = 1$$

(42)

Step C) As per Eq. (27), Sample 2n + 1 denotes the nonlinear function that is used to get the outcome samples.

$$y_{s} = f(x_{s} )$$

(43)

Step D) Compute σ_y and μ_y related to the final variableθ.

$$\mu_{y} = \sum\limits_{s = 1}^{n} {W_{s} \theta_{s} }$$

(44)

$$\sigma_{y} = \sum\limits_{s = 1}^{n} {W_{s} \left( {\theta_{s} - \mu_{y} } \right)} - \left( {\theta_{s} - \mu_{y} } \right)^{T}$$

(45)

The load of the distribution network in this study is considered as a non-deterministic parameter, although its real nature is non-deterministic because the demand in the power network is constantly changing. Evaluating the effect of uncertainty on the exact solution of the problem is very necessary because ignoring it leads to the failure to achieve optimal results. On the other hand, it definitely cannot be said that considering uncertainty weakens or strengthens the goals of the study, which strongly depends on the definition of the input uncertain parameter (network load). But it can definitely be said that the capacity of the parking lots will undergo changes considering the uncertainty of the network load.

The Monte Carlo simulation (MCS) as a conventional stochastic method, typically requires a much larger number of scenarios to accurately model the uncertainty parameter based on the probability distribution function (PDF) compared with the UTM. This extensive scenario generation is computationally expensive and time-consuming. While the significant computational burden can make the MCS less practical for optimization tasks that demand quick decision-making or need to run on limited computational resources. One of the main drawbacks of MCS is the necessity of scenario reduction techniques due to the large number of scenarios it generates. These reduction methods add another layer of complexity and computation to the process. The UTM, however, naturally limits the number of scenarios, eliminating the need for such reductions and further enhancing its computational efficiency. Moreover, the MCS, while highly versatile, may struggle with real-time applications due to its computational intensity. It is more appropriate in scenarios where there is ample time for scenario generation and analysis, or where the focus is on extremely high accuracy rather than computational speed.

Simulation results and discussions

System data

This study used a 33-bus distribution network to implement the suggested EPL optimization framework based on IFHO. There are 33 buses and 32 lines in this network, providing 3.72 MW of active demand and 2.3 MVAr of reactive demand overall. Figure 5 displays the 33-bus network's single-line diagram. Ref.⁴² has also provided information on the buses and network lines of the 33-bus network. For every transmission line, a value is selected over 194.66 h, with two failures annually. Every electric vehicle (EV) is considered to have a 50 kWh battery, and each vehicle's dispatch time is calculated to be 3570 h annually²³. Figures 6, 7 and 8 show the load on a 33-bus network⁴³, the cost of losses⁴⁴, and the price of energy⁴⁵ throughout 24 h, respectively. Each vehicle's annual capital cost is estimated to be 304 $²³, the market price during off-peak hours is chosen to be 0.05 $/ kWh²³, the inverter's efficiency in connecting cars to the grid is estimated to be 85%, and the cost of battery degradation is chosen to be 0.225 $/kWh⁴⁰. Eight EPLs, each with 500 vehicles, have been sited and sized in the 33-bus network according to the suggested technique. The suggested algorithm has identified the EPLs' locations and sizes.

Fig. 5

The 33-bus distribution network.

Fig. 6

Loading changes of 33-bus network during 24 h⁴³.

Fig. 7

Cost of loss changes during 24 h⁴⁴.

Fig. 8

Electricity price changes during 24 h⁴⁵.

The MATLAB software is applied to simulate the suggested technique on a home computer with a Core i7-4510U CPU and a 1 TB HDD. For an accurate comparison, the population, iteration count, and number of independent executions for each algorithm are set to 50, 150, and 25 for the IFHO algorithm, and the algorithm's efficiency is compared to that of the traditional FHO, PSO, GWO, MRFO, and AEFA.

To evaluate the recommended method, simulation scenarios are defined in following.

Scenario#1: EPLs optimization in 33-bus network without BDC and UTM based-NLUn.

Scenario#2: EPLs optimization in 33-bus network with battery degradation cost and without UTM based-NLUn.

Scenario#3: EPLs optimization in 33-bus network with battery degradation cost and UTM based-NLUn.

The objective function of Eq. (1) has five different objectives, and technically some of these objectives are in conflict with each other. To solve the multi-objective problem, the method of weighted coefficients is used as follows to find the optimal weighted coefficients. First, the value of each of the coefficients for different objectives in Eq. (1) is considered equal to 0.2 (fix condition), in other words, the problem is solved. The value of the objective function and each of the five objectives is obtained by satisfying the constraints of the problem. Then the optimization algorithm solves the problem for the entire set of possible weighted coefficients in terms of the following assumptions. a) Any weight coefficient can be changed between the 0 and 1, and b) The minimum and maximum value of the sum of the absolute values of the weighted coefficients of five objectives are equal to 0 and 1, respectively. Finally, the set of weight coefficients that obtains the lowest objective function by satisfying the constraints of the problem is determined as the best coefficients set. If the obtained objective function corresponds to the best coefficients set is lower than the one achieved in equals coefficients conditions, it is determined as the optimal weighted coefficients. Finally, the optimal weighted coefficients set found for Eq. (4), and the best solution (decision variables) as well as the objectives value is computed.

Results of scenario#1

In the first scenario, the optimization results of the EPLs in the 33-bus network are presented via the IFHO without incorporating the BDC of vehicles and also without incorporating the NLUn. The performance of the proposed IFHO to solve scenario 1 has been evaluated with the conventional FHO, PSO, GWO, MRFO, and AEFA algorithms. The weight coefficients of ${\widetilde{\Psi }}^{1}$, ${\widetilde{\Psi }}^{2}$, ${\widetilde{\Psi }}^{3}$, ${\widetilde{\Psi }}^{4}$, and ${\widetilde{\Psi }}^{5}$ in Eq. (1) are obtained 0.27, 0.23, 0.13, 0.18, and 0.19, respectively using the IFHO for scenario 1. The convergence processes achieved from the solution of scenario 1 using different optimizer are presented in Fig. 9. As it is clear, the suggested method has obtained the optimum answer with lower convergence tolerance and better convergence speed. On the other hand, it can be seen that the improvement of the algorithm has increased the optimization accuracy and achieved a better (lower) value of the OF.

Fig. 9

Convergence curve of different optimizer for the scenario#1 solving (a) IFHO (b) FHO (c) PSO (d) GWO (e) MRFO (f) AEFA.

The results of the optimization of EPLs in the network of 33-bus using IFHO and its comparison with other algorithms in scenario 1 are presented in Table 1. According to Table 1, it is concluded that the IFHO has located four EPLs in buses 8, 14, and 29 of the network with the number of 275, 354, and 349 EVs, respectively. In this way, the comparison with other algorithms shows that the IFHO achieves lower loss cost, ENS cost, substation cost, and EPLs cost, as well as a lower amount of network voltage oscillations, which shows the proposed method's superiority in solving scenario 1 solving. The loss cost, ENS cost, grid cost, EPLs cost, and network voltage deviations are achieved at $37,396, $333,347, $999,574, $1,181,920, and 0.0148 p.u, respectively. Also, the statistical analysis of 25 independent runs for different methods for scenario#1 based on Table 2, shows that the IFHO has obtained better values of indicators with less standard deviation in comparison with the traditional FHO, PSO, GWO, MRFO, and AEFA algorithms, so the better performance of the IFHO is confirmed in terms of statistical analysis.

Table 1 The results of EPLs optimization in the 33-bus network (scenario#1).

Table 2 Statistical analysis of different algorithms to solve the scenario#1.

The changes in power losses during 24 h and after the EPLs optimization using the proposed optimizer are shown in Fig. 10. As it can be seen, by determining the size and optimal site of the EPLs in the network, the loss of network lines is reduced by 29.63% in comparison with the base network (((53,144–37,396)/ 53,144)*100) = 29.63%).

Fig. 10

Power losses of 33-bus network using the IBWO for scenario#1.

Based on Fig. 11, it can be found that the voltage oscillations of the network have been reduced. Therefore, the optimal application of the EPLs in the network has improved the voltage profile by 15.91% (((0.0176–0.0148)/ 0.0176)*100) = 15.91%).

Fig. 11

Voltage profile of 33-bus distribution network using the IBWO for scenario#1.

Figure 12 shows the unsupplied energy changes of network subscribers in scenario 1 using the IFHO without and with the EPLs optimization. As can be seen, the optimization of the EPLs in the network has significantly improved the reliability of customers and reduced load interruptions by 71.95% (((1,188,585–333,347)/ 1,188,585)*100) = 71.95%).

Fig. 12

ENS of 33-bus distribution network during a day using the IBWO for scenario#1.

Also, the change curve of the power purchased from the substation in scenario 1 is depicted using the proposed IFHO algorithm in Fig. 13. It is granted that by optimizing the EPLs, the network capacity has been released and the purchased power from the post has decreased in different hours by 41.02% (((1,694,846–999,574)/ 1,694,846)*100) = 41.02%). Also, the Decreasing percentage of different objectives compared with the base network for scenario#1 is demonstrated in Fig. 14. As it is obvious the values of loss cost, ENS cost, grid cost, and EPLs cost, and network voltage deviations are decreased compared to the base network by 29.63%, 15.91%, 71.95%, and 41.02%, respectively.

Fig. 13

Purchased grid power during a day using the IBWO for scenario#1.

Fig. 14

Decreasing percentage of different objectives compared with the base network for scenario#1.

Results of scenario#2

Under the second scenario, the IFHO is used to display the optimization results of the EPLs in the 33-bus network without the UTM-based-NLUn and considering the BDC. The traditional FHO, PSO, GWO, MRFO, and AEFA approaches have been compared to see how well the suggested IFHO performs in solving scenario 2. The weight coefficients of ${\widetilde{\Psi }}^{1}$, ${\widetilde{\Psi }}^{2}$, ${\widetilde{\Psi }}^{3}$, ${\widetilde{\Psi }}^{4}$, and ${\widetilde{\Psi }}^{5}$ in Eq. (1) are obtained 0.30, 0.20, 0.15, 0.17, and 0.18, respectively using the IFHO for scenario 2. Figure 15 displays the convergence processes that were discovered when scenario 2 was solved using several techniques. The suggested approach has produced the best result, which has a faster rate of convergence and a smaller tolerance for convergence. However, algorithmic improvements have led to a better (lower) value for the OF and an increase in optimization accuracy.

Fig. 15

Convergence process of different algorithms to solve the scenario#2 (a) IFHO (b) FHO (c) PSO (d) GWO (e) MRFO (f) AEFA.

Table 3 presents the outcomes of the EPLs optimization in the 33-bus network using IFHO and compares it with alternative algorithms in scenario 2. Table 3 indicates that the IFHO has identified three EPLs with corresponding numbers of 310, 233, 360, and 280 EVs in buses 8, 14, 24, and 31 of the network. This demonstrates the superiority of the suggested algorithm in addressing scenario 2 when compared to alternative algorithms, as the IFHO achieves lower quantities of loss cost, ENS cost, grid cost, and EPLs cost, as well as a smaller amount of network voltage deviations. These are the figures that are extracted for the loss cost, ENS cost, grid cost, EPLs cost, and network voltage deviations: $39,284, $341,193, $1,023,986, $1,230,986, and 0.0152 p.u. Furthermore, as demonstrated by Table 4's statistical analysis of 25 independent runs for each optimizer for scenario#2, the IFHO outperformed the traditional FHO, PSO, GWO, MRFO, and AEFA algorithms in terms of statistical analysis, yielding better indicator values with lower standard deviation.

Table 3 Numerical results of EPLS optimization in 33-bus distribution network (scenario#2).

Table 4 Statistic analysis of different algorithms to solve the scenario#2.

Figure 16 illustrates the decreasing fraction of distinct objectives about the base network for scenario #2. It is evident that the values of network voltage deviations, loss cost, ENS cost, grid cost, and EPLs cost are all lower than those of the basic network by 26.08%, 13.64%, 71.26%, and 39.58%, respectively.

Fig. 16

Decreasing percentage of different objectives compared with the base network for scenario#2.

Results of scenario#3

The results of applying the IFHO, which takes into account both the NLUn based on the UTM and the BDC, to optimize the EPLs in the 33-bus network are illustrated in Scenario #3. The ability of the IFHO to resolve scenario 3 is evaluated to that of the conventional FHO, PSO, GWO, MRFO, and AEFA methods. The weight coefficients of ${\widetilde{\Psi }}^{1}$, ${\widetilde{\Psi }}^{2}$, ${\widetilde{\Psi }}^{3}$, ${\widetilde{\Psi }}^{4}$, and ${\widetilde{\Psi }}^{5}$ in Eq. (1) are obtained 0.31, 0.20, 0.10, 0.18, and 0.21, respectively using the IFHO for scenario 3. The results of IFHO are contrasted with those of other algorithms in Table 5. According to Table 5, the IFHO has located four EPLs in buses 14, 21, 31, and 32 of the network, totaling 125, 172, 427, and 332 EVs. This indicates that the recommended optimizer is better than other algorithms in handling scenario 2 since the IFHO achieves lower amounts of network voltage changes, loss cost, ENS cost, grid cost, and EPLs cost. The numbers that were determined for the network voltage variations, loss cost, ENS cost, grid cost, and EPLs are as follows: $43,796, $361,873, $1,087,923, $1,251,031, and 0.0159 p.u. Furthermore, the IFHO performed better statistically than the traditional FHO, PSO, GWO, MRFO, and AEFA methods, as shown by the better indicator values it obtained with a lower standard deviation, as shown by Table 6 presents the results of the statistical analysis of each optimizer for scenario #3 which the IFHO has obtained better statistical criteria and also lower computational time (CT). Furthermore, concerning the base network, Fig. 17 illustrates the decreasing fraction of distinct objectives for scenario #3. As demonstrated in Fig. 17, the values of the network voltage deviations, loss cost, ENS cost, grid cost, and EPLs cost are all less than those of the base network by 21.06%, 12.15%, 70.82%, and 39.10%, respectively.

Table 5 Numerical results of EPLS optimization in 33-bus distribution network (scenario#3).

Table 6 Statistical analysis of different algorithms to solve the scenario#3.

Fig. 17

Decreasing percentage of different objectives compared with the base network for scenario#3.

Results comparison

In this research, the optimization of the EPLs in the network is implemented in three scenarios. In scenario 1, the EPLs optimization is performed without considering the BDC and without considering NLUn. In the second scenario, the EPLs optimization issue is developed considering the BDC and without considering the UTM based-NLUn, and finally, in the third scenario, the EPLs optimization is done considering the BDC and also the NLUn. In this section, the performance of each scenario in improving network performance has been evaluated.

In all scenarios, the IFHO has achieved a lower OF value in comparison with other methods. The findings have shown that improving the capability of the traditional FHO based on the Taylor-based neighborhood technique (TBNT) has strengthened the IFHO optimization performance in achieving better objective values compared to the conventional FHO (see Fig. 18).

Fig. 18

The OF value obtained by different scenarios and algorithms.

Comparing the performance of different algorithms in solving all three scenarios according to Tables 7, 8, 9, 10, 11, and Figs. 19, 20, 21, 22 and 23, it has shown that the proposed IFHO algorithm compared with FHO, PSO, GWO, MRFO, and AEFA has obtained lower values of annual loss cost, voltage deviation, ENS cost, grid cost, and EPLs cost. Therefore, the suggested methodology linked with the IFHO with the EPLs optimization in the network has been able to achieve the greatest network performance improvement and the lowest value of each of the goals.

Table 7 Results comparison of annual energy loss for different scenarios and algorithms.

Table 8 Results comparison of voltage deviation for different scenarios and algorithms.

Table 9 Results comparison of ENS cost for different scenarios and algorithms.

Table 10 Results comparison of grid cost for different scenarios and algorithms.

Table 11 Results comparison of EPLs cost for different scenarios and algorithms.

Fig. 19

The annual energy loss value obtained by different scenarios and algorithms.

Fig. 20

The voltage deviation value obtained by different scenarios and algorithms.

Fig. 21

The ENS cost value obtained by different scenarios and algorithms.

Fig. 22

The grid cost value obtained by different scenarios and algorithms.

Fig. 23

The EPLs cost value obtained by different scenarios and algorithms.

The results comparison of the EPLs optimization in the 33-bus network in the first and second scenarios has shown that considering the BDC causes undermining various objectives. According to Fig. 24, by incorporating the BDC in the OF, the annual losses, voltage deviations, ENS cost, grid cost, and EPLs have increased by 5.05%, 2.70%, 2.47%, and 2.44%, respectively.

Fig. 24

Increasing the percentage of different objectives using the IFHO considering BDC.

The presented battery–based vehicles model for the parking lots incorporating battery degradation cost prioritizes the impact of the CiL over CaL, aligning with a common focus in battery lifecycle studies. Empirical research has consistently shown that operational factors, particularly the DOD and charging/discharging cycles, play a significant role in battery degradation. Studies clear that higher DOD and cycling accelerate degradation, reducing the overall battery lifespan. Also, the use of SOC constraint to manage degradation also aligns with findings that extreme SOC levels can lead to faster degradation on the battery. Empirical studies often emphasize that battery degradation can vary based on environmental conditions, user behavior, and operational practices. By accounting for these uncertainties, the model mirrors real-world scenarios more accurately, leading to more robust predictions. Moreover, as battery technologies evolve, the costs associated with battery production, replacement, and degradation are likely to decrease. Future models might need to account for these lower costs, which could change the economic calculus of battery usage in the optimization framework. The reduced cost of replacement batteries would impact the long-term battery degradation cost and might lead to more aggressive cycling strategies being economically viable.

Comparing the outcomes of the optimization of EPLs in the second and third scenarios for a 33-bus distribution network reveals that taking the NLUn into account undermines several goals. The annual losses, voltage deviations, ENS cost, grid cost, and EPLs cost have increased by 11.49%, 4.61%, 5.94%, and 1.63%, respectively, when the NLUn based-UT is taken into account (Fig. 25). For distribution network planners to be aware of anticipated faults and achieve correct planning and decision making, NLUn must be taken into account.

Fig. 25

Increasing the percentage of different objectives using the IFHO considering NLUn.

Conclusion

In this paper, to minimize the annual costs of power losses, power purchased from the substation, unsupplied energy of subscribers, and cost of vehicles to the grid, and network voltage deviations while taking BDC and UTM-based NLUn into consideration, a stochastic multi-objective optimization of the EPLs was carried out in the 33-bus network. The novel optimization method known as IFHO improved by utilizing the Taylor-based neighborhood technique and was able to find the best site and capacity of the EPLs in the 33-bus distribution network. Three situations involving BDC and NLUn applications are simulated. The following is how the research's findings were presented:

• The outcomes of scenario#2 for the EPLs optimization in the network considering the BDC and without NLUn demonstrated that incorporating the BDC, the annual losses, voltage deviations, ENS cost, grid cost, and EPLs have increased by 5.05%, 2.70%, 2.47%, and 2.44% compared to the optimization without the BDC.

• The findings of scenario#3 for the EPLs optimization in the network considering the BDC and the NLUn cleared that considering the NLUn based- UTM to model the load uncertainty, the annual losses, voltage deviations, ENS cost, grid cost, and EPLs has increased by 11.49%, 4.61%, 5.94%, and 1.63% compared to the optimization without NLUn.

• The evaluation of the simulations performed in all scenarios has shown the superiority of the proposed IFHO improved using the TBNT in comparison with conventional FHO, PSO, GWO, MRFO, and AEFA algorithms so that the IFHO was able to obtain better objective values and best solution with better convergence process.

• For utility companies, these findings highlight the importance of integrating advanced forecasting and optimization tools that account for load uncertainty to enhance operational efficiency, reduce costs, and support sustainability goals. Policymakers, on the other hand, can use these insights to develop regulations and incentives that promote the adoption of best practices in EPL optimization, drive renewable energy integration, and ensure the long-term reliability and resilience of the electrical distribution network.

• The robust EPL optimization in the distribution network integrated with the multi-energy storage is suggested for future work. The suggested future work involves addressing a range of technical, economic, regulatory, and environmental challenges. Solutions such as advanced modeling, hierarchical control structures, life cycle cost analysis, policy framework development, and stakeholder collaboration will be crucial in overcoming these challenges. By tackling these issues, utility companies and policymakers can unlock the full potential of multi-energy storage, leading to more resilient, efficient, and sustainable electrical distribution networks.