Chance-Constrained Optimization of Photovoltaic System Allocation considering Power Loss, Voltage Level, and Line Current

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Introduction
Nowadays, all countries in the world are making great eforts to cope with global warming along with the need for energy [1].As a result of the high demand for electricity, there is a demand for renewable energy generation and planning studies that take into account the probabilistic issues that come with it.Conventional centralized generation units can aid to some extent in transmitting electricity over large areas due to environmental and cost concerns.Within the scope of these challenges, reducing power losses and keeping bus voltages and line currents within the required limits are becoming crucial issues due to technical and economic reasons [2].In the event that the load voltages and line current fows violate their related bounds, the electricity demand cannot be efciently supplied.Te increase in power losses caused by voltage drops causes voltages to violate their corresponding bounds while transferring energy to meet demand in the electricity distribution network [3].On the other hand, with the widespread penetration of photovoltaic systems, managing the associated uncertainties that pose signifcant risks to system reliability is a major challenge.Tese issues have a negative impact on system security and reliability [4].Besides the production of high-efciency generation technologies, such as high-efciency solar cells for more utilization of solar energy, the need to optimize the reduction of power losses in active distribution networks has led researchers to investigate the optimal arrangement of distributed generation (DG) systems [5][6][7][8][9][10][11][12].In the context of photovoltaic plants integrated distribution systems, it is important to assess the power loss, bus voltage limits, and line current carrying capacities.Tis evaluation becomes more complex due to uncertainties in the solar irradiance and load demand.To address this challenge, electricity grid plans can be developed using a stochastic framework.Tese plans aim to ensure that the necessary energy is supplied to electrical loads.In that manner, planning photovoltaic systems with probabilistic constraints in active distribution networks instead of precise deterministic limits can contribute to more efcient and reliable use of energy.
A number of diferent strategies have been researched and put into practice in an efort to tackle the issue of loss reduction [13][14][15].Te loss reduction software package selection, the issue description, the utilized procedure, and the reached outcome are important criteria for those strategies that vary from one another.Te optimal capacity and allocation of DG units have been handled by taking into account the approach for alleviating distribution power losses and boosting voltage profle in [16].Te decrement in line current fows and the usage of present distribution feeders in case of any loading situations can be observed as the advantages by the optimal distribution power loss minimization.In that manner, the methodologies based on optimal DG placement and size for minimizing losses have been presented in [17,18].On the other hand, the load variation is another crucial issue for planning of the distribution network.As a result of these variations, the operating characteristics of the distribution network fuctuate.Alleviation of power loss and improvement of voltage profle have been taken into account by optimal coordination of DG and voltage regulation devices considering variation of demand in [19].In [20], energy loss minimization has been considered together with the reconfguration of distribution grid under various electricity consumption levels to perform the DG planning.Optimal planning has been carried out for mitigating power losses and managing voltages with the consideration of low, medium, and high loading conditions in the presence of DG allocation in [21].Te power losses have been optimized by taking into account both the DG allocation and capacity in the distribution grids with varying electrical consumption in [22].Te aforementioned approaches disregard the temporal fuctuations of renewable energy and include only single demand or load variation conditions.
Te optimization procedure of loss minimization without considering renewable energy systems might not depict the desired outcomes due to the uncertainties of demand and generation [23,24].In [25], it is shown that the variation of DG penetration level has a signifcant efect on the power loss and voltage profle and it is stated that the penetration is not a static metric.In [26], the optimal renewable unit placements and capacities for alleviating energy losses have been determined using analytical methodology by considering the variations in load and renewable power output under diferent scenarios.In this regard, the minimization of energy loss has been considered with the optimal renewable sites in [27].In [28], a metaheuristic optimization algorithm is applied to reduce energy losses in sizing PV and energy storage system considering uncertainties of demand and generation.Te loss minimization in the planning framework has been dealt with the optimal PV system and voltage regulation device coordination in [29].Te inclusion of optimally located renewable energy sources can be used as an advantage both in terms of reducing power losses and improving voltages [30][31][32][33].When the variability of demand is superimposed on the variability of output power intermittent renewable generation, the optimal placement of these units in the distribution system becomes a challenge optimization problem [34][35][36][37].In [34], it is stated that the scenario-based stochastic optimization method with the inclusion of power loss can lead to higher benefts from renewable-based DG and reduced green gas emission.In [35,36], this problem is optimally considered in the DG-incorporated distribution grids by taking into account the combination of heuristic and stochastic methodologies.However, the chance-constrained approach has not been handled in these studies.In [37], an iterative-based probabilistic approach is presented to minimize average loss with and without consideration of background harmonics.It is clear that power loss minimization plays an extremely signifcant role in the provision of consistent and steady power backup to the customers as a result of the assessment up to this point.However, there is a gap in published research on loss minimization under the framework of stochastic optimization with chance constraints under diferent uncertainty conditions.In light of this, this present study focuses on investigating the impact of uncertainties in PV output power operation under diferent irradiance scenarios on power loss minimization with a chance-constrained probabilistic optimization environment.Te proposed approach has been compared with the existing research in the literature by considering the subjects taken into account in this study in Table 1.As presented in Table 1, the important gaps have been observed in the previous literature based on various criteria.Tese criteria can be classifed by considering the inclusion of metaheuristic optimization methods, load uncertainty, renewable uncertainty, chance-constrained objective function, Monte Carlo Simulation, and diferent solar irradiance scenarios (low, medium, and high irradiances) in the methodologies.Te novelty of this present study is to consider all these criteria in the proposed approach by flling the literature gaps.
In this paper, the uncertainty scenarios are handled with MCS under the GA-based optimization, and the pdf of bus voltages and line current are obtained to be used in chanceconstrained stochastic programming.Te uncertainties of the loads and diferent solar irradiance scenarios are introduced into the GA optimization within an integer programming problem for the optimal allocation of PV plants in the distribution system.In addition, the same methodology is adapted to the DE optimization technique to compare the optimization results.In order to demonstrate efectiveness of the proposed methodology, the obtained optimization results are tested under the set of uncertainty conditions.By combining meta-heuristic optimization and MCS technique under one framework, this paper contributes to the knowledge base of how to allocate PV plants within Te remaining parts of this paper are organized as follows.Section 2 illustrates the load fow analysis methodology contemplated in this study.Section 3 demonstrates the uncertainty modelling taken into account in the present paper.Section 4 gives the heuristic and stochastic featured optimization planning framework proposed in this paper.Section 5 presents the case studies considered in the current study.Section 6 illustrates the results and discussion drawn from the proposed approach.Finally, Section 7 provides the general conclusions determined in this paper.

Load Flow Analysis
Te incorporation of renewable PV systems in distribution grids remains prominent as a result of rising electrical loads.In that manner, the bus voltages, line current fows, and losses can be considered while making use of the PVs in these grids.At the same time, the variability of electrical demands and solar radiation-dependent PV power outputs can have an impact on the technical parameters of the distribution network.Te analysis of load fow is indispensable for examining these parameters in the grid.In the present article, the load fow analysis, which is based on backward and forward sweep [38], has been considered to examine the parameters of the grid.Tis approach has been demonstrated as follows: where DV iter+1

B
is the vector demonstrating the iteratively changes of bus voltages, I iter B,inj is the vector illustrating the injection currents associated with PVs and electrical loads at the present iteration, V init B is the vector illustrating the initial bus voltages, V iter+1 B is the vector depicting the bus voltages at another iteration, BIBC is the matrix illustrating the relations between bus and line currents, and BCBV is the matrix demonstrating the relations between line currents and bus voltages, respectively.

Uncertainty Modelling
In this article, the beta distribution with a stochastic structure has been used to generate solar radiation conditions [39].Te expression of this distribution is demonstrated as follows: where where Srad cr b is the radiation considered for the PV system at bus b of distribution system for the cr th state.Te variables of beta expression, which are α and β, can be obtained in terms of deviation (σ) and mean (μ) as follows:

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Te injected PV output powers can be determined with respect to the radiation as follows: where P PV b is the installed power of renewable PV system.Tis power will be optimally integrated at bus b of distribution system in the planning.
In the present research, the states of electrical demand have been produced by considering the expression of normal distribution [39] as illustrated in the following: where σ Dem,b illustrates electrical load standard deviation, P Dem,b,avrg demonstrates the mean of load consumption, and P cr Dem,b depicts the load power at bus b of the distribution network for the cr th state.

Planning Framework
Te chance constraints have been considered while optimally minimizing the expected power losses.GA [40] methodology has been carried out with the integer framework in the distribution network.Te optimization problem has been dealt with probabilistic structure considering variabilities of electrical load and PV output powers.Te bus voltages and line currents have been regarded as the chance constraints.Te GA methodology jointly with MCS approach has been implemented for optimal execution of power loss minimization.
where Tcr is the total amount of states for uncertainty in the planning phase, TNB is the total bus number in the distribution energy grid, |V cr b | and θ cr b are the voltage absolute and angle values of bus b in the cr th state of planning, |Y be | and δ be are the admittance absolute and angle values in the b th row and e th column of the related matrix, and E(P Loss ) is the expected value of power losses.Te objective function has been subjected to the chance constraints as follows: where V min b and V max b are the minimum and maximum bounds of voltage at bus b, I cr nl is the line fow current value for the line between buses b and e at the cr th state, I max nl is the maximum bound of line fow current, λ V and λ I are levels of confdence corresponding to the bus voltages and line fow currents, and TNL depicts the total line numbers in the distribution network.
By considering the limit violation of constraints, the objective function has been converted to the augmented structure by handling the method of the penalty function. where where ψ b V and ψ nl I illustrate the limit violation probabilities for bus voltages and line currents and ck V and ck I demonstrate the factors for penalizing the constraints in case of violation in this study.Te limits of constraints are sustained while considering the minimization of total distribution grid power losses from the viewpoint of (9).Te factors for penalizing the constraints are performed in case the limit violation is observed [41].Te more proper outcome is optimally investigated by GA when the violation of constraints gives rise to the bigger values in augmented structure.
Tis study implements the chance-constrained optimization to determine optimal allocation of PV systems in distribution network considering power loss, voltage level, and line current under stochastic programming framework.In this study, the decision variables are the optimal bus allocations for the predefned PV system output powers.Te optimal PV placement buses in distribution energy grid are depicted as follows: International Transactions on Electrical Energy Systems where where x PV b is the bus number for the optimally allocated PV system, x PV,min and x PV,max are the minimum and maximum values for the buses, at which PVs are optimally allocated, and TPV represents the total PV system allocation buses.Te block diagram of the proposed methodology is demonstrated in Figure 1.
Te framework of the stochastic optimization process is depicted in Figure 2.
Te prominent phases of the planning methodology for optimally allocating the PVs are demonstrated as follows: (1) Enter the distribution network parameters.
(2) Enter the installed PV power outputs that will be allocated.
(3) Enter the states of solar irradiance and electrical demands produced from the probabilistic distribution functions.(4) Enter the beginning population that demonstrates the PV allocation buses in a random manner.(5) Initiate MCS approach considering the cr th state (cr � 1, ..., Tcr).Carry out phase 8, in case cr is bigger than Tcr.(6) Extract the electricity demand and calculate the solar irradiance-dependent PV output powers for the cr th state.(7) Calculate the bus voltages and line currents on the distribution network by carrying out load fow approach considering the cr th state.(8) Plot the functions of stochastic distributions for bus voltages and line currents on the distribution network.(9) In this phase, the functions of stochastic distributions for the voltages and line currents are considered in terms of limit violation.Te integrals of functions of stochastic distributions are evaluated on the ranges, where the bounds are surpassed [42].An illustrative example of a probability distribution function for the quantifcation of chance constraints in the distribution network is shown in Figure 3.As shown in Figure 3(a), B is the shaded pdf region where the confdence level of bus voltage is maintained.However A and C are pdf regions where the bus voltage limits of the distribution network parameters are exceeded.While the shaded region D in Figure 3(b) can be the pdf region where the confdence level of line current is maintained, E is the pdf region in which the line current constraint is violated.(10) At this stage, the constraints on the fulflment of the technical limitations are checked from a probabilistic point of view according to the confdence level specifed.If the pdf integrals, where the bounds of bus voltages and line currents are violated, are not greater than 1 − λ V and 1 − λ I , these stochastic constraints are maintained.(11) Handle the chance constraints associated with bus voltages and line currents.Tis step is performed while determining the objective function by considering every chromosome.(12) In case the constraints are not maintained, utilize the factor for penalizing the constraints in obtaining the value of objective function.(13) In case the criteria of optimally planning process are maintained, the optimal PV allocations are printed out.If they are not maintained, return to phase 4. ( 14) In terms of examining the constraints for limit violation, implement the testing of optimal PV allocation.
In the optimization process, the PV installed capacity to be placed in the distribution network is entered.Te states of solar irradiance and electrical power demands are generated from probability distribution functions.Ten, the initial population of PV bus allocation is randomly initialized by the GA framework at the upper level of the problem.At the lower level of the problem, probability distribution functions for voltages and line currents in the distribution network are determined with the help of load fow under MCS, and then probabilities of constraint violations are obtained.During the GA process, PV bus allocations are iteratively generated according to the objective function in which the violation cases are included.
In order to demonstrate efectiveness of the proposed methodology, the obtained optimization results are tested under the set of uncertainty conditions.Te implementation of testing of optimal PV allocation outcomes is demonstrated in Figure 4.

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In the verifcation phase of the obtained optimization results, the optimal PV system allocations and distribution network parameters are frst input to the test algorithm.Ten, MCS is applied using the uncertainty sets generated for electricity consumption and solar irradiance and the network constraints are evaluated in terms of limit violation probabilities.

Case Studies
In the proposed stochastic programming, the optimization results are compared for possible long-term solar irradiance conditions under three diferent profles as low, medium, and high.Te frequency distributions of solar irradiances for these scenarios are shown in Figure 5.

Results and Discussion
In the optimization process, the size of population, maximum iteration, rate of crossover, and objective function tolerance have been considered as 30, 200, 0.8, and 10 −6 for GA.Te stochastic optimization has been implemented on the PC possessing the CPU of 2.80 GHz.

IEEE 33 Bus Distribution
Network.Te planning simulations have been carried out on the 33 bus distribution network [43], which is demonstrated in Figure 6.
Te electricity demand states of distribution network are illustrated in Figure 7. Te standard deviation of electricity demands is 10% in terms of the mean of these demands.
Te MCS stage of the optimization procedure has been carried out with 1000 scenarios according to the probability distributions of solar irradiance and electrical load.Te total number of PV systems has been considered to be 12 in the test network.Te installed power of each PV system has been regarded as 400 kW.Te minimum and maximum values for the buses, at which PVs will be optimally allocated, are 2 and 33, respectively.Te level of confdence for the bus voltages and line fow currents is taken as 0.8 in the chance-constrained programming.

Results of GA for Single Run of Stochastic Optimization
Framework.Te GA methodology jointly with MCS approach has been performed for scenarios 1-3.Te optimal PV allocations are demonstrated in Table 2.
As seen in Table 2, the optimal PV system allocations demonstrate variability among the scenarios.It is worthy to note that these optimal PV allocations are the decision variables that have been determined by minimizing the power losses and meeting the chance constraints in all scenarios.Tese PV placements among the scenarios have been obtained by taking into account diferent limitation impacts of chance constraints on the objective function values.Terefore, the chance constraints and various solar radiation profles in the scenarios have important role in the optimal PV allocations and power losses.Te optimally evaluated distribution grid power losses are shown in Figure 8 in an iterative manner.
In the base case, PV systems are not allocated and only variations in electricity demands are considered and the total power loss in the grid is 211.31 kW.By using stochastic planning, the power losses have been cut down to 80.69 kW in scenario 1, 70.38 kW in scenario 2, and 66.65 kW in scenario 3. Te percentages of decrements in these losses have been achieved at 61.81%, 66.69%, and 68.46% for scenarios 1-3 when compared with the base case.Te total power loss in scenario 1 is greater than that in the other scenarios due to the limitations imposed by the chance constraints associated with the bus voltages.In scenario 1, the optimal PV allocation has led to higher power loss when compared to the other scenarios since the voltage profle is less due to the lower solar radiation conditions in this scenario.Te comparatively less voltage profle has given rise to the limitation impact by the chance constraints.As the solar radiation levels get higher in other scenarios, the voltage profle also has the improvement in the distribution system.Tis improvement provides the reduction in power losses and chance constraint limitations.In scenarios 2 and 3, the total grid losses are alleviated more than those in scenario 1 as the levels of solar irradiance rise and the impact of chance constraints reduces in these scenarios.

Comparison between GA and DE for Single Run of Stochastic Optimization
Framework.Te proposed optimization issue in all scenarios has also been handled by the DE methodology for comparing the optimization results obtained in the GA.In this paper, DE methodology has been modifed to handle the integer variables since the proposed problem has discrete framework.Tis modifcation has been achieved by implementing the approach presented in [44].Te optimal PV system placements, corresponding power losses, convergence iterations, and simulation times are presented in Table 3. Te optimally obtained power losses are iteratively demonstrated in Figure 9.
As illustrated in Table 3 and Figure 9, the advantages of GA-based algorithm when compared to DE have been seen.Te optimal power losses have been obtained as 81.8917 kW, 70.994 kW, and 67.0895 kW in the corresponding scenarios by using DE algorithm.Te number of iterations for the convergence is 102, 132, and 147 by the GA algorithm, whereas that is 200 with the DE approach in scenarios 1, 2, and 3, respectively.Moreover, the simulation times are 935.36,1246.11, and 1363.53 seconds for the GA, while those are 4424.73,4366.11, and 4470.92seconds by the DE algorithm.Terefore, the stochastic power loss minimization by optimal PV allocation can be remarkably dealt with by the GA-based algorithm thanks to the aforementioned advantages.In addition, the optimal power losses obtained by the corresponding PV placements have been alleviated by the infuences of solar radiation levels and chance constraint limitations in scenarios 1 through 3 for both algorithms.

Comparison between GA and DE for 10 Runs of Stochastic Optimization
Framework.Te 10 independent runs of both GA and DE optimization algorithms have been considered to compare the algorithm performances.Tese 8 International Transactions on Electrical Energy Systems independent runs have been performed for all scenarios.Table 4 illustrates the optimal power loss, PV allocation buses, iterations, and simulation times for the corresponding 10 runs in all scenarios.
As seen in Table 4, the proposed GA methodology has the advantages in comparison with DE.Te improvements have been generally observed by the aid of GA approach in terms of the power losses, corresponding PV allocation buses, iterations, and elapsed times.Te GA and DE runs, at which the best optimization results have been obtained, are shown with the bold values in Table 4. Te best GA outcome has been seen at 8th run of GA in scenario 1, and the best results have been observed at 3rd runs of GA in scenarios 2 and 3. Te best DE optimization results have been determined at 2nd, 10th, and 3rd runs of DE in scenarios 1, 2, and 3, respectively.Te optimal power losses and PV allocations have been determined by both GA and DE in all scenarios.Te power losses have been reduced by the rising solar radiation in scenarios 1 through 3.
Te determined best optimal power losses among 10 runs are presented in Figure 10 in an iterative manner.As shown in Figure 10, the proposed GA approach's superiority has been observed in comparison with DE.With the GA algorithm, the power losses have been determined as 80.6008 kW,         11 for all scenarios.Te PV placements have converged to their corresponding optimal buses as seen in Figure 11.It is clearly shown in Figure 11 that the best optimal PV placements have been achieved at the diferent sites by applying GA and DE with stochastic optimization framework on the distribution system.For both methodologies, these best PV allocations have been obtained so that the power losses have been minimized and the chance constraints have been satisfed.Tese best placements have been infuenced by the chance constraint and solar radiation levels in the scenarios.Tis in turn has also resulted in the changes of power losses among the scenarios.
Te statistical analysis of algorithms' performance has been conducted for observing the executions of GA and DE methodologies.Te corresponding analysis is presented in Table 5.As seen in Table 5, the outcomes have been given based on 10 independent runs of optimization algorithms.Te improvements have been determined in best, average,    International Transactions on Electrical Energy Systems and worst values of objective function by implementing the GA method in all scenarios.Tese fndings illustrate that the proposed GA approach has presented more improved behavior in comparison with DE.Moreover, the GA methodology has given smaller standard deviation values than DE in all scenarios.Te near-optimal outcome represents the one with the lower standard deviation [45,46].In that manner, the GA-based optimization is more successful than DE since it has the smaller standard deviation.

Validation of Optimization Results
. In this study, the robustness of the best results obtained from the optimization results using GA and DE with 1000 samples in each of three diferent scenarios is tested under the uncertainty sets considering low, medium, and high irradiance cases.In the test results obtained using the load fow under MCS, it is observed whether the bus voltages and line currents will violate the chance constraints.Table 6 shows the violation probabilities of bus voltages in all scenarios for both GA and DE.
As can be seen in Table 6, the boundary violation in the chance constraints of bus voltages is observed in the base case.In all scenarios, including the base case, no violations of the chance constraints of line currents were observed.In scenarios 1-3, the best optimal PV placements for both algorithms were able to signifcantly reduce the violations.Te violation of the chance constraints for bus voltages was achieved at the desired confdence level with the optimal PV   International Transactions on Electrical Energy Systems allocations for each bus of the distribution network.Te confdence level of 0.8 in scenario 1 is almost maintained.Moreover, in scenarios 2 and 3, confdence levels of 0.9 and 0.95 were achieved, respectively.In Figure 12, the voltage at bus 33 and the line current between buses 1 and 2 are presented as an example of the cumulative distribution functions for the base case and scenarios 1 to 3. Bus 33 is prominent in terms of being the far end bus in the distribution grid.Te line between buses 1 and 2 is also important with regard to feeding this grid.By the best optimal PV allocation, the improvements in technical parameters have been observed compared to the base case as seen in Figure 12.Te increase in solar irradiance gives rise to the enhancements in cumulative distribution functions.It is clear from Figure 12 that the cumulative distributions for bus voltages have a shift to the right hand side, whereas those for line currents possess the shift to the left hand side in scenarios 1 through 3 when compared with the base case.Tese shifts demonstrate the improvements in bus voltages and line currents.
Variations in electrical demand and PV systems have a signifcant impact on the constraints of power network.

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When distribution networks with renewable energy sources are planned to incorporate the efects of these probabilistic behaviors, it may be possible to maximize the expected benefts.Te chance constraint approach is an efective method for incorporating this random behavior into the optimization problem.As a result, the problem of minimizing power losses in distribution networks by considering the uncertainties arising from photovoltaic systems together with the uncertainties of the load is of great importance in terms of the most efcient use of available energy.

IEEE 69 Bus Distribution
Network.Te planning simulations have been performed on the 69 bus distribution network [43], which is demonstrated in Figure 13.Te distribution system electricity demand states are presented in Figure 14.For this distribution system, the standard deviation of electricity demands is 10% in terms of mean of these electricity demands.
Based on the probability distributions of solar irradiance and electrical loads, the MCS stage of the optimization methodology has been performed with 1000 states in all scenarios.Te total number of PVs has been regarded as 12 in the distribution system.Te installed power of each PV system has been considered as 400 kW.Te minimum and maximum values for the buses, where PV systems will be optimally placed, are 2 and 69, respectively.Te confdence level for the bus voltages and line currents is considered as 0.8 in the chance-constrained programming.

Comparison between GA and DE for Single Run of Stochastic Optimization Framework.
For scenarios 1-3, the GA optimization methodology together with MCS has been 16 International Transactions on Electrical Energy Systems carried out.Te optimization problem has also been implemented by the DE methodology for comparing the results determined in the GA approach.Te optimal PV system allocations, expected power losses, convergence iterations, and simulation times are presented in Table 7.
For both methodologies, the variabilities of optimal PV system placements among the scenarios are illustrated as seen in Table 7.It is worthwhile mentioning that these optimal PV bus allocations have been obtained by minimizing the power losses and meeting the chance constraints in all scenarios.Te optimal PV placements and power losses can be afected by the limitation impact of chance constraints based on diferent solar irradiance scenarios.Te optimal distribution network power losses are iteratively demonstrated in Figure 15.
For the base case, PV systems are not placed and only variations in electricity loads are considered and the total power loss in the distribution network is 191.7302kW.By applying the GA methodology, the power losses have been decreased to 79.8442 kW in scenario 1, 73.2466 kW in scenario 2, and 69.1757 kW in scenario 3, respectively.Tese power losses have been decreased by 58.36%, 61.80%, and 63.92% for scenarios 1-3 in comparison with the base case.Due to the limitations imposed by the chance constraints related with the bus voltages, the total power loss in scenario 1 is greater than that in the other scenarios.Since the voltage level is less due to the lower solar irradiance conditions in scenario 1, higher power loss has been obtained by the optimal PV placement in this scenario when compared with the other scenarios.Te limitation efect of chance constraints is increased by the comparatively less voltage level in scenario 1. Te distribution system voltage profle is improved when the solar irradiance levels increase in scenarios 2 and 3. Te power losses and chance constraint limitations are minimized by this improvement.Since the solar irradiance levels increase and the impact of chance constraints decreases in scenarios 2 and 3, the total distribution network power losses are minimized in these scenarios more than those in scenario 1.
Te advantages of GA optimization methodology have been observed in comparison with DE as presented in Table 7 and Figure 15.Te optimal power losses have been determined as 80.1766 kW, 73.9686 kW, and 69.5418 kW in scenarios 1-3 by using the DE approach.Te number of iterations for the convergence is 110, 124, and 140 by GA methodology, while that is 200 for DE in scenarios 1-3, respectively.In addition, the simulation times are 2131.394,2402.663, and 2712.684seconds for GA, while these are International Transactions on Electrical Energy Systems 9251.708,9129.139, and 9348.287seconds by the DE approach.As a result, GA-based methodology can handle the stochastic power loss minimization in both less simulation times and less number of iterations when compared with the DE approach.In addition, the total power losses determined by the GA approach have been improved more than those obtained by DE.For both algorithms, the optimal power losses have been minimized in scenarios 1 through 3.

Comparison between GA and DE for 10 Runs of Stochastic Optimization
Framework.Te 10 runs of both GA and DE optimization methodologies have been taken into account for comparing the algorithm performances.For all scenarios, these 10 independent runs have been implemented.Te expected optimal power losses, PV allocation buses, iterations, and simulation times are presented in Table 8 by considering the 10 runs in all scenarios.
Te proposed GA approach has the advantages when compared to DE as shown in Table 8.With regard to the power losses, PV allocation buses, iterations, and simulation times, the improvements have been generally determined by GA methodology.Te GA and DE runs, where the best optimization outcomes have been observed, are shown with the bold values in Table 8.Te best GA optimization results have been obtained at 9th run of GA in scenario 1, 5th run of GA in scenario 2, and 7th run of GA in scenario 3, respectively.Te best DE optimization results have been determined at 10th runs of DE in scenarios 1 and 2, and the best result has been seen at 4th run of DE in scenario 3.In scenarios 1 through 3, the power losses have been reduced by the increasing solar irradiance for both algorithms.
Te obtained best expected optimal power losses among 10 runs are iteratively shown in Figure 16.Te proposed GA methodology's advantages have been determined when compared to DE as presented in Figure 16.Te best optimal power losses have been obtained as 79.8339 kW, 72.2470 kW, and 68.8709 kW taking into account GA approach for scenarios 1, 2, and 3, respectively.However, the best optimal power losses have been determined as 79.9527 kW, 73.0363 kW, and 69.2797 kW for all scenarios using the DE methodology.Te proposed GA methodology converges to the best optimal results at 105, 126, and 120 iterations, whereas the DE approach converges to the best optimal solutions at 200 iterations for scenarios 1, 2, and 3, respectively.Moreover, the GA optimization methodology presents the best optimal results at 2034.513, 2441.415, and 2325.157seconds, while the DE approach gives the best optimal solutions at 9267.212, 9226.335, and 8977.042seconds.Terefore, the proposed GA optimization methodology has the superiority for the best optimal solutions in comparison with DE.
Te best optimal PV system allocations for both GA and DE methodologies are presented in Figure 17 for all scenarios in an iterative manner.As seen in Figure 17, the best PV allocations have converged to their corresponding buses.Te best optimal PV system allocations have been determined at various buses by implementing GA and DE with stochastic optimization framework on the distribution network.Tese optimal PV placements have been achieved so that the power losses have been alleviated and the chance constraints have been maintained.In this paper, the simulation studies have been conducted in the direction of stochastic optimization approach for strengthening the distribution network with the execution of useful and optimal PV allocations.Tese allocations may be utilized for the appropriate supervision of distribution grid.Te entire grid performance can be upgraded with optimally placing the PV units by delivering the demands with the help of lines.Te various uncertainties of load consumption and PV output powers cause the distribution system to have a complex nature.Hence, the distribution grid parameters can have variabilities.In this concept, the optimal PV system allocation provides the power fow delivery to the demands through the lines.Te power loss minimization of entire distribution network can be achieved by this power delivery.With the aim of minimizing the power losses, the optimal PV placements have been properly managed for supplying the lines and demands.Terefore, specifying the allocations of PV systems is inevitable from the standpoint of distribution system loss.
Te statistical analysis of algorithms' performance is given in Table 9. Te results have been presented by considering 10 independent runs of GA and DE optimization algorithms as shown in Table 9. Te improvements have been obtained in best, average, and worst values of optimal power losses by applying GA methodology for all scenarios.In addition, the GA approach has presented lower standard deviation values than DE for all scenarios.It is clearly seen that the proposed GA method has presented more successful results when compared to DE.

Validation of Optimization Results
. Te validation of best optimization results determined from GA and DE optimization frameworks has been performed under the uncertainty sets considering low, medium, and high irradiance scenarios for presenting the robustness of optimal results.Te violations for the chance constraints have been examined for the bus voltages and line currents by taking into account the test results determined with the MCS-based load fow.Te violation probabilities of bus voltages are presented in Table 10 by considering all scenarios for both GA and DE.
Te violations in chance constraints for bus voltages have been observed in the base case as presented in Table 10.Te chance constraint violations for the line currents have not been observed for all scenarios.Te violations related with bus voltages have been considerably minimized by the optimal PV system allocations in scenarios 1-3.For both algorithms, the chance constraint violations for bus voltages have been determined at the desired confdence level by considering the optimal PV system placements in the distribution system.For scenario 1, 0.85 confdence level has been maintained.In addition, 0.95 and 0.99 confdence levels have been achieved for scenarios 2 and 3, respectively.
Te cumulative distribution functions for the voltage at bus 65 and the line current between buses 1 and 2 have been illustrated as the example in Figure 18 for the base case and scenarios 1 through 3. Bus 65 is important with regard to being one of far end buses in the distribution system.Te line connecting buses 1 and 2 is the feeder providing power fow in the distribution network.As shown in Figure 18, the improvements in bus voltages and line currents have been observed when compared with the base case by taking into account the optimal PV placements in scenarios 1 through 3. Te cumulative distribution functions have been improved by the rise in solar irradiance conditions.It is clearly seen in  Te difculty in planning studies in active distribution networks arises from the intermittent characteristics of the output power of photovoltaic systems with increasing participation rates.A more reliable network can be achieved by considering a large number of possible situations in planning studies.A larger number of samples lead to increased computational difculty and even to overly conservative decisions or unfeasibility.On the other hand, planning based on worst-case scenarios with low probabilistic probability of realization results in oversized investments.In this manner, the stochastic planning has important role in dealing with the wide range of samples by taking into account the respective probabilities.In this paper, the probabilistic optimization analysis of distribution network has been provided by the uncertainties of solar radiation levels and distribution system load consumption.Hence, the distribution grid technical parameters can be stochastically handled while determining the optimal control variables.In the current study, the optimal PV allocations have gained great importance in terms of considering the probabilistic sample ranges of uncertainties, handling the chance constraints of technical parameters with the specifed confdence levels, and minimizing the power losses of distribution system under diferent scenarios.

Conclusions
In this study, power losses, voltages, and line capacities are taken by the chance constraint optimization method in the planning of distribution networks under uncertainties of load and solar irradiance, and the problem of optimal placement of photovoltaic systems, whose penetration level in the grid is increasing day by day, is solved by combining meta-heuristic algorithms such as GA and DE and Monte Carlo Simulation.In the proposed stochastic programming, the optimization results are compared for possible long-term solar irradiance conditions under three diferent profles as low, medium, and high.According to the simulation results, distribution grid power losses are optimally minimized in these three scenarios, compared to the scenario with load change but without PV connection.Te power loss in the low irradiance scenario is higher than that in the other two scenarios due to the limiting efect of chance constraints related to bus voltages.In the medium and high irradiance scenarios, it is observed that the power loss can be further reduced as the solar irradiance levels are high and the efects of chance constraints are reduced.Te fndings show that in distribution networks where PV systems with diferent irradiance uncertainty conditions are connected, violations of the grid voltage are particularly important.At this point, the optimization decisions obtained with the probabilistic grid constraint approach were able to provide quantifable results.Furthermore, the comparison between MCSembedded GA and DE algorithms has been made for illustrating the efectiveness of the proposed methodology.Te simulation outcomes show that the proposed stochastic GA approach has the superiority.Both GA and DE converge to the optimal solutions.Nevertheless, GA performs in less computational time and iteration in comparison with DE.

Figure 8 :
Figure 8: Te optimally evaluated distribution network power losses.

Figure 9 :
Figure 9: Iterations of algorithms in case of a single run in the 33 bus distribution system.

Figure 10 :
Figure 10: Best power losses in algorithms for 10 runs of stochastic optimization framework in the 33 bus distribution system.

Figure 11 :
Figure 11: PV allocation convergence curves for; (a) scenario 1 in GA, (b) scenario 1 in DE, (c) scenario 2 in GA, (d) scenario 2 in DE, (e) scenario 3 in GA, (f ) scenario 3 in DE for the 33 bus distribution system.

Figure 12 :
Figure 12: Te cumulative distribution functions of (a) bus voltage for GA, (b) line current for GA, (c) bus voltage for DE, (d) line current for DE in the 33 bus distribution system.

Figure 15 :
Figure 15: Iterations of algorithms in case of a single run in the 69 bus distribution system.

Figure 16 :Figure 17 :
Figure 16: Best power losses in algorithms for 10 runs of stochastic optimization framework in the 69 bus distribution system.

Figure 17 :
Figure 17: PV allocation convergence curves for; (a) scenario 1 in GA, (b) scenario 1 in DE, (c) scenario 2 in GA, (d) scenario 2 in DE, (e) scenario 3 in GA, (f ) scenario 3 in DE for the 69 bus distribution system.

Table 1 :
Comparison of the proposed approach with the existing research in the literature.

Table 2 :
Te optimal PV system allocations.

Table 3 :
Results of algorithms for single run of stochastic optimization framework in the 33 bus distribution system.

Table 4 :
Results of algorithms for 10 runs of stochastic optimization framework in the 33 bus distribution system.

0079 33, 15, 8, 30, 31, 28, 7, 24, , 18, 25, 12 914.902
45,ts at 4303.45, 4412.59, and 4416.60 seconds.As a result, the GA-based methodology has also advantages in the best optimal results when compared to DE.Te evaluated best PV system allocations among 10 runs of both GA and DE algorithms are iteratively shown in Figure

Table 5 :
Statistical analysis for GA and DE algorithms in the 33 bus distribution system.

Table 6 :
Te probabilities of limit violations for GA and DE in the 33 bus distribution system.

Table 7 :
Results of algorithms for single run of stochastic optimization framework in the 69 bus distribution system.

Table 8 :
Results of algorithms for 10 runs of stochastic optimization framework in the 69 bus distribution system.

Table 12 :
[43] 69 bus distribution network data[43].International Transactions on Electrical Energy Systems cumulative distribution functions that the right hand side shifts have been obtained in bus voltages and the left hand side shifts have been determined in line currents for scenarios 1 through 3 in comparison with the base case.It can be inferred from these shifts that the bus voltages and line currents have been improved.