Analysis of Pre-and during COVID-19 Mixed Load Models on Unbalanced Radial Distribution System Using a New Metaphor-Less Rao Optimization

. An unbalanced electrical distribution system (DS) with radial construction and passive nature sufers from signifcant power loss. Te unstable load demand and poor voltage profle resulted from insufcient reactive power in the DS. Tis research implements a unique Rao algorithm without metaphors for the optimal allocation of multiple distributed generation (DG) and distribution static compensators (DSTATCOM). For the appropriate sizing and placement of the device, the active power loss, reactive power loss, minimum value of voltage, and voltage stability index are evaluated as a multiobjective optimization to assess the device’s impact on the 25-bus unbalanced radial distribution system. Various load models, including residential, commercial, industrial, battery charging, and other dispersed loads, were integrated to develop a mixed load model for examining electrical distribution systems. Te impact of unpredictable loading conditions resulting from the COVID-19 pandemic lockdown on DS is examined. Te investigation studied the role of DG and DSTATCOM (DGDST) penetration in the electrical distribution system for variations in diferent load types and demand oscillations under the critical emergency conditions of COVID-19. Te simulation results produced for the mixed load model during the COVID-19 scenario demonstrate the proposed method’s efcacy with distinct cases of DG and DSTATCOM allocation by lowering power loss with an enhanced voltage profle to create a robust and fexible distribution network.


Introduction
Te amount of electrical energy consumers require is snowballing, spurred by new electrical and electronic appliances.All categories of the consumer class, such as residential, commercial, and industrial participated in energy growth due to the growth in electric vehicles and other intelligent devices based on the consumption of electrical energy.In addition to being a necessary part of society, the electrical power distribution industry's job is critical, because it is dependent on upstream power resources and needs to provide an adequate supply to all classes of customers with high-quality power without interruptions.Te electricity grid is rapidly growing, delivering reliable, and afordable electricity to everyone by predicting the level of load demand, which changes and fuctuates continuously.Te COVID-19 pandemic continues to have a signifcant impact on the world's energy systems.In this scenario, the modest decline in energy demand caused by direct limitations on industry, commerce, and other activities, and the general economic slump have disproportionately afected the power sector [1].Tis article analysed the infuence of the COVID-19 pandemic lockdown on demand, operation, and supply in the Indian power system [2].In addition, as shown in Figure 1, an unusual situation in load demand seen due to the COVID-19 epidemic has signifcantly impacted the energy consumption and trajectory of India.As a result, load demand scenarios are critical for comprehensive knowledge of the distribution network and system operation, planning, and long-term strategy.Tis article suggests the signifcant penetration of DG into the distribution network for the development of defence mechanisms meant to improve the grid's resiliency against COVID-19-type incidents [3].
Te main components of the power system are the power generation plant, transmission system, and distribution system (DS).Tey form the value chain of power delivery to the consumer [4].DS is the most important part of the value chain because it is exposed to failure and disturbance caused by the consumer side as well as the generation and transmission side.Te power loss in the DS is much higher than in the transmission system because of the extreme R/X ratio and a major portion of investment is allocated to the DS of the total electrical industry.Te voltage profle of DS is afected by load uncertainty due to the highly nonlinear characteristics of the load.Te poor voltage stability could lead to a total shutdown of the DS.Te reactive power unbalance in the DS severely afects its performance while delivering power to the consumer.A large fuctuation in reactive power introduces harmonics that will degrade the quality of power and the chances of voltage collapse will increase.Load demand volatility may be addressed by providing sufcient assistance through DG and DSTAT-COM (DGDST).
Integration of DGDST changes DS's characteristics from passive to active with multidirectional power fow [5].It will make DS ill-conditioned by having negative impacts.Tese include reverse power fow and unaccepted voltage levels, combined with high active power loss (APL) and reactive power loss (RPL) if the DGDST location and size are nonoptimal.Terefore, an optimal solution consisting of the location and size of DGDST obliges us to overcome the negative impacts and achieve the positive efects by maintaining necessary constraints within their tolerable limits for practical load demand.Integration of DGDST makes the DS more resilient to any unhealthy situation on the grid.It also enhances the efciency of DS by depreciating the power loss and increasing the voltage stability.
DS can be categorized based on diferent structures defned in the power system: radial balance distribution system (RDS), radial unbalanced distribution system (RUDS), mesh distribution system, etc. Te DS is an intrinsically unbalanced system due to serving single and three-phase loads via a distribution transformer.Te loads in each phase are unequal.Distribution lines are not transposed like transmission lines.Tere are already many researchers analyzing RDS to enhance the performance of DS by means of integration of DGDST [6].Generally, the constant load is taken for the analysis of DS, but practically, diferent types of load are served by DS, including residential load, industrial load, and commercial load; so there is a need for a mixed load demand that consists of all types of load with a suitable participation factor [7].In the presented work, RUDS have been considered for the integration of DGDST individuals and simultaneously to enhance the performance of a new mixed load model.DGs are powergenerating units usually integrated into DS to decrease power loss and maximize the voltage profle to ensure secure and reliable delivery.It is becoming popular because it avoids the need for new distribution lines near the end user and prevents new transmission lines in the middle part of the value chain.Tat makes it an economical and adopted solution for DS [8].Nowadays, DGs with renewable energy 2 International Transactions on Electrical Energy Systems sources, like PV and wind-based DGs, etc., strengthen the concept of DG installation due to its environmental concern by decreasing carbon emissions [8].Mostly, loads are inductive in nature, which causes an unbalance in reactive power.DSTATCOM is the promising solution for steadystate reactive power compensation in DS by absorbing both active and reactive power by inserting a voltage of inconstant magnitude and phase angle at the point of coupling connection in DS.It will maintain the voltage within permissible limits and increase the stability of the system [9].Since DGs with a nonunity power factor can also inject necessary reactive power into DS for compensation of unbalancing of reactive power, it will be an uneconomical solution because of the higher cost of DG and minimum regulation in voltage compared with the DSTATCOM support system.Optimal allocation of DGDST can provide several technical and economic advantages.It minimizes the APL, and RPL and maximizes the voltage stability.Te improved voltage profle (IVP), lower branch current, improved power quality, and increased trustworthiness of the DS attained afterward integration of DGDST [10].
Very little research has been described for the planning of DGDST in RUDS.Te power loss is lessened by the placement of DG using the voltage index method and sizing computed by the variation technique algorithm.Te size should not be greater than 20% of the feeder loading in RUDS with voltage-dependent LM; it is a combination of constant power, constant current, and constant impedance loads [10].To compensate for reactive power, a multishunt capacitor is allocated optimally using hybrid particle swarm optimization considering harmonics.Te problem was developed as a nonlinear integer programming problem with inequality constraints [11].Optimal capacitor placement is executed using a simulated annealing technique, including a greedy search technique to make a balance between the quality of solution and computational speed in a substantial practical scale distribution system considering diferent load level peaks [12].Reverse power fow constraints are incorporated for the optimal location of solar-based DGs in practical RUDS.It gives criteria for limiting the maximum optimal size of DG.Te size of DG and DSTATCOM should be according to the loading at each phase.An equal rating of DGDST is not preferable as it increases reactive power loading and energy saving.Te optimal size and location of both DG and KVAR support are determined using particle swarm optimization (PSO) by minimizing losses in RUDS [13].Te voltage profle is better when an unequal size of KVAR support is provided in each phase compared to the rating of the same size.Various metaheuristic techniques (GA, PSO, and BSFLA-PRTPLF) were implemented to allocate biomass-based DG in RUDS, including a probabilistic load model to enhance the voltage profle with desirable voltage limits on all load buses [14].Simultaneously optimization for phase balancing by providing the required complex power and size of conductor for RUDS using a diferential evolution (DE) algorithm results in signifcant improvement in power loss and a drop in voltage [15].
Because of the recent COVID-19 pandemic, energy demand has been lowered drastically.Tis pandemic increased demand for residential loads due to changed habits around the world, as people typically stay home and work from home, if possible, although there is a signifcant decrease in commercial and industrial loads because governments worldwide have been forced to limit business activity in response to reducing the threat of coronavirus [16].Tis catastrophic situation poses new challenges for the technological and fnancial operations of the power sector [17].Technically, the distribution industry sufered from negative and positive load growth subsequent to voltage infringement and was economically weak due to a substantial decline in demand for industrial and commercial loads despite having surplus electricity generation from renewable and nonrenewable sources [18].Terefore, most electrical utilities worldwide have launched an emergency recovery initiative to resolve these current problems and risks.Te changed situation during COVID-19 created a greater need for DG allocation because upstream electricity from traditional power plants was disturbed.Terefore, the purpose of this analysis includes the investigation into the DG allocation considering scenarios during COVID-19, accompanied by the never-seen load growth challenges confronted by EDS.As the impact of COVID-19 increased, all countries worldwide imposed a lockdown.Te electrical industry was also afected due to the lockdown by the shutdown of factories and commercial activities, so the power demand decreased.However, the generation of electricity remained at peak demand since electricity storage is not possible, so this imbalance in power became large and uneconomical because most of the revenue comes from prime consumers (industrial and commercial).Many governments have imposed a "lockdown" on their citizens to reduce communal spread, which afects the energy sector.In this context, this paper [19] examines COVID-19's impact on the global energy market, especially in India, and describes the operation of diferent countries and how they secured their power sector throughout the pandemic.In a lockdown, people stayed at home, so residential demand increased.It created a change in LM, which caused a steep fall in load demand, and created a stability problem, which made it a challenging task to operate DS [20].
COVID-19's crisis and lockdown constraints have lowered activities and energy use.Commercial and public administration operations have increased energy consumption, and the residential sector has gained scale economies [21].For the COVID-19 situation, a new LM equation was adopted due to changes in diferent loading share types in active and reactive power loading.In critical situations like lockdown due to COVID-19, the demand varies according to load type.Residential demand increases, whereas commercial and industrial demand decreases, and battery charge loading and other loading are also afected proportionally.All the changes made in the load weighting factor are assumed by analyzing the report on the electrical industry's frst response to COVID-19 for diferent electrical systems [22].Te peak load demand of the Indian power sector is varied, as shown in Figure 2. Te fuctuation in peak International Transactions on Electrical Energy Systems load demand causes energy fuctuation of the power network depicted in Figure 3. Te paper gives ofcials a more detailed idea of the pandemic's infuence on electricity demand to minimize losses [23].
From the literature review, it has been found that the optimal allocation problem of DG as well as DSTATCOM is solved by various researchers using both deterministic and stochastic approaches using sensitivity analysis and metaheuristic optimization algorithms [24,25].In the optimization feld, there are so many algorithms based on the metaphor of the behavior of animals, fshes, insects, or any natural phenomenon.It is ambiguous to select one algorithm for an optimization problem by tuning the decision variable with an algorithm-specifc parameter to get the best result.To overcome this issue, a new metaphor-less Rao optimization algorithm is taken for minimizing the objective function [26].A proposed optimization algorithm has been employed in the multiobjective optimization of selected thermodynamic cycles [27].Te multiobjective function is formulated by assigning suitable weight to every objective function based on the priority of DS [28].Te problem of DSTATCOM allocation is given less attention compared to DG allocation for RUDS since a reactive power support capacitor is installed in the DS, but with the invention of power electronics devices, DSTATCOM is a better option for mitigating issues related to reactive power imbalance.Simultaneous allocation of DGDST is very rare for RUDS, whereas in BRDS many studies have been found for single and multiple allocations of DGDST [29].In [30,31], DGs are modelled as PV nodes delivering real power at unity power factor are taken for URDN analysis.Although DSTAT-COM's primary role is to provide reactive power (as required) to the PCC-modelled voltage control system, it is modelled as a constant source of reactive power to provide adequate support for reactive power compensation [32,33].
During COVID-19, electric utilities implemented various sorts of resilience thinking towards power system resilience through decision-making processes [34].
Installation of DGDST will convert DS into a smart distribution system (SDS) for unpredicted fuctuations in load demand caused by any emergency situation like the COVID-19 pandemic.
Te literature review suggests the following research gap: (i) Te load demand from a diferent class of consumers plays a vital role in determining the existing EDS performance by incorporating DG.Most of the research work followed a constant power load model for DS analysis.However, little work has been carried out to modify the practical LM based on residential, commercial, and industrial load types only.In the proposed work, LM is composed by aggregating the load demand raised by the diferent classes of consumers, including battery charges and other residual loads, to analyze the DS that was not considered before.(ii) From the literature, it is learned that metaheuristic optimization provides more accurate and reliable outcomes with lower computational time than sensitivity-based and classical optimization methods due to certain advantages.Te optimal allocation of DG in EDS is a nonlinear optimization problem, and there is continuous scope for improving the optimal solution with fewer complicated approach in a minimum time with reasonable accuracy.(iii) COVID-19 worldwide afected the energy sector.
Te EDS was afected due to the lockdown declared, resulting in an imbalance between supply and demand from various LM.From the literature, it is revealed that DG installation enhances the capability of EDS to meet the increased demand for the load.Terefore, it is vital to analyze the consequences of COVID-19 for DG allocation designed for positive and negative growth in load demand by various LM.International Transactions on Electrical Energy Systems Te main contribution to this work can be summarized as follows: (i) Te efect of COVID-19 is taken to analyze the DS and its impact is balanced by DGDST allocation in lockdown duration (ii) A new mixed load model has been formulated for performance analysis of RUDS (iii) A multiobjective function consists of APL, RPL, MVV, and VSI, which is considered a minimization problem (iv) Simultaneous allocation of multiple DG and DSTATCOM in URDS is performed separately following successive allocation of one, two, and three DG and DSTATCOM to minimize multiobjective function (v) A new Rao metaheuristic optimization free from a metaphor-based hypothesis is proposed to resolve the enigma of DGDST allocation Te remaining paper is organized as follows.In Sections 2-4 the detailed structure of the problem formulation is given, including objective function and load fow analysis.Te load fow for the test system is provided in Section 5. Section 6 presents an overview of the Rao optimization algorithm, followed by applying the proposed algorithm to the DGDST optimal allocation problem.Simulation results were discussed briefy in Section 7. Eventually, the conclusion and future scope of the proposed work are given in Section 8.

Load Models (LMs)
Earlier, most of the research work was carried out with a constant load model, but DS cannot be subjected to a single load model characteristic that will optimize the system for all types of installation of DGDST.Te load deterministic modelled according to diferent LMs by using exponential load components for active and reactive power load values is shown in Table 1 [35].Te weightage of load components is taken from [28,36].In this work, along with the constant load model, a new mixed load model is considered for the optimization problem of DGDST allocation.In many cases, the load demand characteristics of the distribution system are determined by the constant load.

Constant LMs.
A mathematical equation has been developed for load fow analysis to compute load demand according to the load type.Furthermore, the optimal allocation of DG is carried out based on the characteristics of each load.Te loads are not constant in actuality, so all types of loads, residential, commercial, industrial, battery charge, and remaining loads are considered.Te remaining load types include all the ignored loads that are not covered by any type of load to cover the maximum span of loading.Tat will make the load equation more realistic for analysis of DS in a realistic manner, like agriculture load.Te constant load model is formulated as a function of nominal voltage and bus voltage raised by the power exponent given by the following equations: where P is the active power load, P Oj nominal active power load, Q is the reactive power load, Q Oj is the nominal reactive power load, V j is the operating voltage for the constant load, V o is the nominal voltage, and C po and C qo are the load exponential coefcient of active and reactive power for the constant load, respectively It is a voltage-independent model.Any change in voltage magnitude will not afect system demand, so it remains constant irrespective of any change in voltage.International Transactions on Electrical Energy Systems 2.2.Mixed LMs.In practical situations, loads are not constant, since DS is dependent on consumer mix, providing power to diferent types of load according to their active and reactive demand, so loads are a combination of unique types of load demand, for instance, residential, industrial, and commercial [37].Te mixed load is composed by taking diferent weightage of all types of load.Te historical trend of diferent classes of consumers in India is utilized to formulate a new mix load demand model equation.Te weighting components-based load modelling is adopted.Mixed load analysis is also carried out using a diferent weighting factor for each load depending upon the impact.Te weighting factor is decided by the consumption of active and reactive power by the various types of load given in Table 2, so total active and reactive demand is distributed as follows: where P MIX and Q MIX are active and reactive power for the mixed load, respectively, and K R , K CM , K I , K B , and K OT are the weighting factors for residential, commercial, industrial, battery charge, and the other load, respectively.Te mixed load model is voltage dependent.Any deviation in voltage will be caused by specifc types of load exponents that will give corresponding active and reactive power demand.Tat will change the steady state condition of the DS.Terefore, the mixed model will give an actual estimate of all the DS parameters for load fow analysis and further investigation.

LM during COVID-19
Te coronavirus has triggered widespread devastation; a major casualty is the power sector.During the COVID-19 pandemic, many countries imposed a nationwide lockdown.Electrical DS was afected by it because of the shutdown of factories and industries.Tere was a huge plunge in electrical demand from industrial and commercial users, whereas due to people staying in homes, the residential demand was high [30].Te demand growth is decreased by the lockdown and makes the distribution sector vulnerable.Te power shared by residential consumers is higher than industrial, commercial, and other types of consumers [38].A major portion of revenue comes from industrial and commercial customers for DISCOM.A reduction in demand could lead to a huge loss of revenue.For the COVID-19 situation, new load model equations have been adopted due to changes in diferent types of loading share in active and reactive power loading to analyze practical load uncertainty.In this present work, the impact of this situation is shown by modifying the weighting component of individual load and further fnding a new solution for optimal allocation of DGDST during the lockdown.Table 2 presented the value of the weighting factor [22].

Problem Interpretation
In this section, problem formulation and load fow are addressed.An objective function has been developed, which will be adopted for the optimal allocation of DGDST in optimization problems relating to the proposed optimization algorithm.Figure 4 presented the general description of an unbalanced distribution system.

Active Power Loss (APL).
In DS, the R/X ratio is high and, due to its radial construction, there is a huge loss of active power.To augment the capability of DS, the primary objective is to reduce its APL, which will maximize the performance of DS.Te APL is determined using the branch current loss formula given by the following equations: where P a Loss,br(xy) , P b Loss,br(xy) , and P c Loss,br(xy) represent the active power losses in distinct branches for phase "a", "b," So the frst objective function is subjected to minimization of the total APL of the DS. (5)

Reactive Power Loss (RPL).
Te large amount of RPL in the DS causes instability and fuctuation in voltage, which results in system collapse.For the smooth operation of the DS, the reactive power loss (RPL) should be minimized.Tat will make the DS more resilient to any unhealthy situation.Te reactive power loss is determined using the branch current loss formula given by the following equations: where Q a Loss,br(xy) , Q b Loss,br(xy) , and Q c Loss,br(xy) represent the reactive power losses in distinct branches for phase "a", "b," and "c," respectively.Te total RPL is determined by the summation of all phase losses of the distributed line, given by equation (5).
Hence, the second objective function is subjected to minimization of total RPL of the DS.

Minimum Value of Voltage (MVV).
It is the minimum value of voltage (MVV) among all the buses in the DS.Te MVV is important for the system stability of the DS.Te lower magnitude of MVV indicates a high probability of system reliability, whereas its value close to the reference value indicates the better reliability of DS with less probability of system collapse.
where MVV a , MVV b , and MVV c represent the MVV in distinct branches for phase "a", "b," and "c," respectively.Te total RPL is determined by the summation of all phase losses of the distributed line, given by the following equation: So the third objective function is subjected to maximization of the minimum voltage magnitude of the distribution system.

Voltage Stability Index (VSI).
It is defned as the capability of the DS to maintain the voltage within a permissible range.Te zero value of VSI indicates the voltage collapse, whereas it becomes unity for a healthy DS.It is calculated by the following equation.
where VSI a y ,VSI b y ,andVSI c y are phasewise voltage stability index of receiving bus y.P y , and Q y are active and reactive power of bus y.R xy , and X xy are resistance and reactance between buses x and y.
Te VSI is determined for each phase and the minimum value of the VSI is summed up for maximization.International Transactions on Electrical Energy Systems Hence, the fourth objective function is subjected to maximization of the minimum voltage stability index magnitude of the DS.
4.5.Overall Objective Function.Te objective function of the problem is to fnd the optimal location and sizing of DG and DSTATCOM individually and together that will reduce the APL and RPL and maximize the MVV and VSI.Te single objective function optimization problem is converted to a multi-objective function using suitable weighting factors for every single objective depending upon priority is formulated as where w is the weighting factor.Now, the minimization of the multiobjective function is an optimization problem.

System Constraints.
Tese constraints are checked in every iteration for the feasible solution according to the requirements of the DG and DSTATCOM allocation optimization problems.Te developed multiobjective function is subjected to the following equality and inequality constraints as follows.
4.6.1.Equality Constraints.Te algebraic sum of the power in the electrical distribution network and loss should be identical to the power delivered by the DG and DSTAT-COM.Since the nonoptimal allocation of DG and DSTATCOM could afect the operation of DS, that condition is restricted by applying the following constraint: where P D active power demand raised by the load, Nl total number of distribution lines, NDG number of DG, P DG active power injected by DG, Q D reactive power demand raised by the load, Q DST reactive power injected by DSTATCOM.

Inequality Constraints
(1) Voltage Magnitude Constraint.Te voltage magnitude after each iteration should be within the tolerance limit; otherwise, it will lead to a problem of unbalancing in voltage magnitudes, since there is a large fuctuation in bus voltage, and then it will inject a higher amount of current through the branches.Tat would make the solution nonoptimal.
(2) DG Size Constraint.Te power injected by DG should be within the described limits so that we can operate DS consequently.Te power supplied by DG is controlled by real power constraints.
(3) DSTATCOM Size Constraint.Te power supplied through DSTATCOM should be within reasonable limits for the efcient functioning of DS.Te power supplied by it is controlled by reactive power constraints.
(4) Line Loading Constraint.Te line loading should not be greater than the maximum permissible limit, and it creates a line outage due to excessive power fow.In this work, several types of load are taken, so it is worth considering line loading constraints to fnd the optimal solution.
where S max is the maximum loading on the line connected between two buses, it is the maximum power transfer capability of that line.

Load Flow with DG and DSTATCOM
Te load fow method for RUDS shown in Figure 5 is occupied from [39].It is composed of the multiplication of two matrices.Te frst is the bus injection to a branch-current matrix (BIBC) and the other is the branch current to bus voltage matrix (BCBV).Te relationship calculations are given for phase a as follows: where I a and B a are the branch currents and equivalent current injection vector for phase a, ∆V a is the diference between bus voltage and substation voltage for phase a. Bus voltage is a function of current fowing in branches and the voltage of DS along with network parameters.Now, from equations ( 21) and (22), DLF a is the direct load fow matrix for phase a. Te relation between bus voltage and equivalent current injections of buses for phase a can be given as In RUDS, there will be three conductors, so all the equations are similarly derived for phases b and c.Te solution to load fow is found by using the following iterative formulas.
I k a,i is equivalent current injection for phase a, ith bus during the kth repetition.V k a,i is the voltage of phase a, ith bus in the kth repetition.P i,a + jQ i,a is complex power for phase a, ith bus is determined as follows: P iD,a + jQ iD,a is the total demand of the DS for phase a. P iG,a + jQ iG,a is the total generation for DS for phase a.
A DG with unity power factor (UPF) is taken as a source to provide adequate active power into DS, whereas DSTATCOM is modelled as a source to provide reactive power.Te change in complex power will modify the current of "equation ( 24)." where s is the size of DG, l is the location of DG, and d is the dimension of DG d � 1, 2, 3.
where c is a conversion coefcient, PF DG power factor of DG unit.
In this work DG with unity power factor (UPF) PF DG � 1 p.u.
Te new modifying current with DG is given by the following equation: Q i,a,DSTATCOM is the reactive power provided by DSTAT-COM, it is given by [34], according to the following equation: DSTATCOM's transformed voltage and inserted reactive power are used for continued forward sweep to evaluate load currents in the next iteration of backward sweep load fow.Te iterative formulas will be modifed according to the voltage magnitude injected by DSTAT-COM as followed:- where I k i,a,DST is the updated current after integration of DSTATCOM in the DS.Now, with the installation of DGDST, the new iterative current is modifed according to the following equation.
where I k i,a,DGDST is the updated current after integration of DG and DSTATCOM in the DS.If the power factor of DG is unity, then the modifed term is given by "equation (27)."International Transactions on Electrical Energy Systems Te numerator of equation ( 30) is decreased owing to DG's active power and reactive power injected through DSTATCOM.Te denominator increases due to the injection of DSTATCOM voltage, hence leading to an increase in the voltage of DS and lower power loss.

Rao Algorithm
Tere are various metaphor-based optimization algorithms for optimization.However, from the efectiveness point of view, all techniques are not perfect or give an unreliable solution; the computational time is also high.Tat is why, to solve complex problems with accuracy, there is a necessity for metaphor-less optimization algorithms [26].It is an uncomplicated concept without the fne-tuning of artifcial particles that will make implementation more straightforward than conventional metaphor-based techniques.Te algorithm was successfully implemented for multiobjective engineering optimization, which showed promising results [27].
Te benefts of the Rao algorithm are as follows: (1) It uses no metaphor for the optimization of multiobjective problems, which makes it superior (2) In this algorithm, the best and worst solutions amid optimization of a given problem are obtained, and interactions occur randomly between the candidate solutions (3) It necessitates general control parameters like population size and iteration numbers; no other algorithm-specifc parameters are involved (4) It has already been tested on various dimensions' benchmark functions with constrained and multimodeling properties, which proves the attractiveness of adopting this algorithm for diferent optimization problems First of all, initialize the initial population to minimize the OF.If m number of decision variables are there and n number of possible solutions for any iteration i.In the optimization process, let the best agent attain the OF's best value in the whole search space with all possible solutions.Similarly, the worst agent obtains the OF's worst value.
If X j, k, i is the value of decision variable X for the jth variable during the ith iteration for the kth agent.Te magnitude is going to change according to the following equations: where X j,best,i is the value of the variable j for the best candidate and X j,worst,i is the variable j for the worst candidate during the ithiteration.X' j,k,i is the updated value of X j,k,i and r1, j,i is random number for the j th variable during the i th iteration in the range [0, 1].Te value of the decision variable is compared with a randomly selected agent solution.According to the ftness function, information is exchanged according to equations ( 35)- (37).
Te decision variable value will be updated according to the ftness of the OF.If the previous solution was superior to the current solution, then the agents' value would be replaced with a better OF value in the current solution.If the current solution is more promising than the previous solution, then the agent's value will be replaced by the previous solution.So the loop will continue until convergence criteria are not attained; it may be the maximum number of iterations or any other tolerance or mismatching criteria according to the problem.In this paper, the Rao algorithm is implemented on the DG optimization problem with 50 populations and 100 iterations.Convergence characteristics are plotted for various LMs that show the best, worst, and average results, respectively, after several independent runs.Te fow chart is given in Figure 6.Te convergence characteristics for formulated problems are drawn for different distinct cases shown in Figure 7.
Te optimal value of the objective function is obtained after 100 runs.Te results with minimum objective values are selected for the convergence curve under three diferent settings of the proposed algorithm with 50, 100, and 150 maximum iterations.After 100 runs, the optimum objective function value is determined.Under diferent settings of the proposed method, the outcomes with the least objective values are chosen for the convergence curve.
Te Rao algorithm improves the result by considering the diference between the best and worst solutions.Te new value of a variable is obtained by adding the diference between the best and worst values of a variable multiplied by a randomly generated number during the iteration.Te algorithm improves the result by considering the diference between the best and worst solutions and considering the candidate solutions' random interactions.In the Rao algorithm, all the accepted function values at the end of the iteration are maintained.Tese values become the input to International Transactions on Electrical Energy Systems  International Transactions on Electrical Energy Systems the next iteration.Te randomly generated number r1 used to multiply the diference between the best and worst solutions helps in the excellent exploration of the search space.Te absolute value of the variable for the candidate is considered in equation (35); furthermore, equation (37) enhances the algorithm's exploration ability.Te randomly generated number r2 is used to multiply the diference between a variable's values corresponding to a candidate and another randomly selected candidate.Tis helps in the acceptable exploitation of the search space.For these reasons, the Rao algorithm has been proven to have better competitive results.

Simulation Results
Te proposed Rao optimization algorithm is tested on a standard three-phase IEEE-25 RUDS shown in Figure 8. Te base value of system voltage and power is 4.16 kV and 30 MVA respectively.Te IEEE-25 bus system contains 25 buses connected with 24 branches.From [40] the line and load data of the test system is taken.Te investigation into the tested system was carried out under the following conditions for fnding the optimal size and location of DGDST using a formulated multiobjective function: (i) Mixed load model demand Pre COVID-19 efect (ii) Mixed load model demand during COVID-19 efect For the purpose of assessing objective function, the optimum placement is performed using a single, two, or three DG, DSTATCOM, or both DG and DSTATCOM.Te size of DGDST is maintained within a specifc range.Te lowest range is zero, and the maximum range is equal to the loading on each phase.All instances are evaluated using setting 3 of the suggested method in this load due to its superior performance in covering all the possibilities and efectively utilizing the exploration and exploitation of the optimization algorithm.Te simulation results are described in the following section.3 includes the optimal allocation results phase-wise for DG and DSTATCOM for all the cases, pre-COVID-19 efect.Te global minima of multiobjective functions are obtained for the values of size and location of DGDST for each case.For single DG and DSTATCOM individual allocation problems, bus location 7 was found optimal, whereas for single DG and DSTATCOM together, bus location 7 was found optimal for DG and bus location 9 was found optimal for DSTATCOM.It is revealed from the table that a particular bus is found optimal for all the phases because diferent bus locations for diferent phases introduced phase unbalancing into the unbalanced distribution system.

Mixed LMs. Table
Table 4 summarizes the phase-wise outcome of all stages for all cases after allocation of DG and DSTATCOM, pre-COVID-19 efect.Te value of APL and RPL decreased, whereas the MVV and VSI magnitude increased.Te APL is 40.759kW for phase a, 43.192 kW for phase b, and 33.583 kW for phase c in the basic scenario, which is decreased to 19.83 kW, 20.989 kW, and 16.627 kW for PhABC in Case1, respectively.Furthermore, it decreased to  International Transactions on Electrical Energy Systems 712.78 (7) 626.93 (7) Case 2 1DSTATCOM 406.21 (7) 434.23 (7) 436.08 (7) Case 3 1DGDST 951.32 (7) 1002 (7) 420.05 (7) 357.97 (9) 494.96 (9) 468.76 (9) Case 4 Te RPL is also decreased for PhABC to 15.593 kVar, 16.76 kVar, and 15.491 kVar from 45.442 kVar, 41.399 kVar, and 44.872 kVar in the primary case.For PhABC, the MVV was increased from 0.93881 p.u., 0.93827 p.u., and 0.94491 p.u. to 0.96995 p.u., 0.96876 p.u., and 0.97153 p.u.In addition, the VSI was increased for PhABC from 0.77368 p.u., 0.77122 p.u., and 0.79378 p.u. to 0.88492 p.u., 0.87676 p.u., and 0.8874 p.u.Following DG, DSTATCOM is positioned ideally inside the distribution system to maximize performance.After repeated allocation of DSTATCOM, the APL for PhABC decreased to 31.555 kVar, 31.734kVar, and 27.718 kVar, respectively.In addition, the RPL for PhABC is reduced to 35.72 kVar, 31.804kVar, and 31.101kVar, respectively, compared to the original base case result.In addition, PhABC optimizes the MVV to 0.96091 p.u., 0.97032 p.u., and 0.96202 p.u. from 0.84928 p.u., 0.88242 p.u., and 0.85298 p.u. are the improved VSI values for PhABC.It was concluded from the fndings that although DSTATCOM installation decreases APL and RPL and increases MVV and VSI, it exhibits less improvement in objective function than DG placement, implying that concurrent deployment of DG and DSTATCOM would enhance outcomes in a more stressful situation.Terefore, it is revealed that the allocation of DGDST is a better approach compared to the individual allocation of DG and DSTATCOM.
Te voltage profle is shown in Figure 9 for all cases considered.Te voltage profle enhanced signifcantly from the base case to sequential cases.It is clear from the voltage profle that the voltage of all the phases is within tolerance limits.However, the 3DGDST allocation (Case 9) provides the best voltage profle compared to other cases.
From the above-given results, it is clear that the obtained optimal values provided an excellent outcome by reducing the APL and RPL values as well as maximizing the MVV and VSI values for the unbalanced distribution system for the pre-COVID-19 scenario.Te same approach to the allocation of DG and DSATCOM individually and together will be applied to mixed load model demand during COVID-19 in the next section.

Mixed LMs during COVID-19 Efect.
In COVID-19, demand is infuenced by the fuctuations in load demand according to diferent classes of consumers with positive and negative load growth described in section 2 (COVID-19 efect).
Table 5 summarizes the optimal results for DG and DSTATCOM allocation.Te location and size obtained for all the cases were diferent compared to the pre-COVID-19 LM due to unpredicted load demand oscillation.International Transactions on Electrical Energy Systems Table 6 summarizes the results followed by the optimal allocation of DG and DSTATCOM for various formed cases.A similar trend is observed in COVID-19 load demand after optimal allocation of DG and DSTATCOM from Cases 1 to 9. Te best result was obtained for Case 9, like the previous pre-COVID-19 scenario.
In the basic scenario, the APLs for phases a, b, and c are 40.987kW, 43.44 kW, and 33.753 kW, respectively, which are lessened to 13.802 kW, 15.559 kW, and 9.4786 kW for PhABC following multiple DGDST allocations in Case 9.In addition, RPL is decreased to 12.258 kVar, 13.849 kVar, and 11.708 kVar for PhABC in Case 9, down from 45.683 kVar, 41.641 kVar, and 45.118 kVar in the base case.
It is evident again from the fndings for Cases 1 to Case 9 that although DSTATCOM installation reduces APL and RPL and increases MVV and VSI, it exhibits less improvement in objective function than DG placement, implying that concurrent deployment of DG and DSTATCOM would enhance outcomes in a more vital COVID-19 situation.
Te voltage profle is shown in Figure 10 for all the considered cases.As can be seen from the graph, under the mixed load model, demand improvement is enhanced from Case 1 to Case 9. Again, the combination of 3DGDST produced the best voltage profle compared to other cases.

Comparative Analysis of Pre and during COVID-19
Situations.From Figures 7, 9, and 10, it is clear that the optimal allocation of DG and DSTATCOM by the Rao algorithm improves the performance of the distribution network.However, the 3DGDST (Case 9) found an optimal solution because of maximum improvement and the equivalent resolution in both pre-and during the COVID-19 situation, making the distribution grid more resilient to emergency situations caused by highly unpredictable nonuniform load demand scenarios.Figure 11 presented the comparative graphs for all the cases of mixed LM pre-and during COVID-19 efect.

Conclusion
In the presented work, the optimal allocation of DGDST is done using the Rao optimization algorithm with mixed load models.Te efect of COVID-19 was analysed by taking into account the variation in load demand during the lockdown period.Te optimal value of single, two, and three DG, DSTACOM, and combined DGDST are evaluated for minimization of active and reactive power loss, and maximization of MVV and VSI considering a multiobjective function.Te OF is minimized when DG is placed.Te addition of DSTATCOM also gives better results, but when simultaneously DGDST is sited at optimal size, the OF is minimized by enhancing voltage profle, MVV, VSI, and reducing APL and RPL.Tis makes the distribution system more reliable and resilient to any critical load demand scenario.Te size and location of DGDST during COVID-19 confrms the validity of the proposed approach using Rao optimization.Te integration of DGDST improved the performance of the unbalanced radial distribution system for any fuctuation in the load demand scenario.Finally, all results show that maximum improvement in DS can be achieved when 3DGDST is allocated simultaneously compared to other cases.
Te allocation of DG varies according to the load type and demand, so this study will help the DS operator to select a DGDST combination of the desired rating at the proper location for mixed load demand through the proposed innovative approach.Te present work can be extended by using time interdependence modelling in combination with various voltage-dependent static and dynamic load models headed for optimal allocation of renewable sources-based DG.COVID-19 has disturbed the DS fnancially and technically, suggesting researchers for optimal planning of the EDS with the negative rise in load demand that industrial and commercial users have encountered for the frst time in the history of the electricity era.International Transactions on Electrical Energy Systems

Figure 2 :
Figure 2: Peak demand variation of India.

Figure 4 :
Figure 4: General description of unbalanced distribution system.

Figure 8 :
Figure 8: Single line layout of 25 bus unbalanced radial distribution system.

Table 1 :
Te value of exponential factor.

Table 2 :
Te value of weighting factor.

Table 3 :
Te optimal size and location of DG and DSTATCOM for mixed LM pre-COVID-19 efect.

Table 4 :
Te phase wise result analysis for mixed LM pre COVID-19 efect.

Table 5 :
Te optimal size and location of DG and DSTATCOM for mixed LM during COVID-19 efect.

Table 6 :
Te phase wise result analysis for mixed LM during COVID-19 efect.