Scenario-Based Network Reconﬁguration and Renewable Energy Resources Integration in Large-Scale Distribution Systems Considering Parameters Uncertainty

: Renewable energy integration has been recently promoted by many countries as a cleaner alternative to fossil fuels. In many research works, the optimal allocation of distributed generations (DGs) has been modeled mathematically as a DG injecting power without considering its intermittent nature. In this work, a novel probabilistic bilevel multi-objective nonlinear programming optimization problem is formulated to maximize the penetration of renewable distributed generations via distribution network reconﬁguration while ensuring the thermal line and voltage limits. Moreover, solar, wind, and load uncertainties are considered in this paper to provide a more realistic mathematical programming model for the optimization problem under study. Case studies are conducted on the 16-, 59-, 69-, 83-, 415-, and 880-node distribution networks, where the 59- and 83-node distribution networks are real distribution networks in Cairo and Taiwan, respectively. The obtained results validate the effectiveness of the proposed optimization approach in maximizing the hosting capacity of DGs and power loss reduction by greater than 17% and 74%, respectively, for the studied distribution networks.


Introduction
Wide integration of renewable energies has been promoted in many countries [1,2]. The decentralized allocation of renewable distributed generations (DGs) has been supported by distribution system operators (DSOs) to cover loads at peak loading durations, reduce the loading on distribution systems' transformers, and also minimize the total active loss; however, this led to many operational problems, including line thermal limits overloading, overvoltage violations, harmonic overloading, and others. Thus, the requirements for optimal DGs planning has become essential to attain lower operational issues.
up to the 4400-node distribution networks via Cuda programming implementation for the DNR optimization problem.
In this work, a multi-objective bilevel optimization approach is employed to assess the maximum HC and also maximize the total power loss reduction of six distribution networks up to the 880-node large distribution network. The optimization process is conducted by setting two optimization levels, i.e., the upper and the lower levels, to address the planning, and operation viewpoints, simultaneously. In the upper level, two multi-objective optimization techniques, including multi-objective non-dominated sorting genetic algorithm (NSGA-II) [24] and multi-objective particle swarm optimization (MOPSO) [25] are employed to maximize two objectives, including HC maximization of the distribution networks and maximization of the total power loss reduction. In the lower level, the graphically based DNR algorithm in Reference [18] takes place to maximize the power loss reduction of the distribution network at each scenario.
In this work, the main contributions are listed as follows: (1). Optimal allocation of WTs and PVs is performed while considering their uncertainties, including wind speed and solar irradiance, respectively. (2). Load uncertainty is considered in this study to step on the real-life benefits of DGs penetration during load alterations. (3). A bilevel multi-objective optimization approach is formulated to optimally size renewable WT/PV DGs along with network optimization from the planning and operational perspectives. Further, the optimal solution is chosen from the pareto solutions via a decision-making algorithm called 'Technique for Order of Preference by Similarity to Ideal Solution' (TOPSIS). (4). Case studies are conducted on real distribution networks, including the 59-node distribution network in Cairo and the 83-node distribution network of the Taiwan power company. Furthermore, the proposed optimization approach is tested on the 415-and 880-node large distribution networks, which are ensembled from the 83-node real distribution network.
The organization of this work is enclosed in five sections. Section 2 provides the problem statement of this work, which is composed of the power flow equations, the DNR graphically based algorithm, the WT, and PV modeling, the used scenarios generation algorithm, and TOPSIS. Section 3 formulates the bilevel multi-objective optimization approach used in this paper. Section 4 encloses the results and commentary on the obtained results. Finally, Section 5 briefly illustrates the research outcomes and future recommendations.

Materials and Methods
In this section, the power flow equations, the graphically based DNR mathematical algorithm, load, WT, PV DGs' models, TOPSIS, and system performance indices are illustrated in detail. Figure 1 demonstrates distribution network modeling for the upcoming case studies. and 136-node distribution networks. In Reference [23], a very time-efficient optimization approach was proposed by Roberge et al. to optimize large distribution networks up to the 4400-node distribution networks via Cuda programming implementation for the DNR optimization problem. In this work, a multi-objective bilevel optimization approach is employed to assess the maximum HC and also maximize the total power loss reduction of six distribution networks up to the 880-node large distribution network. The optimization process is conducted by setting two optimization levels, i.e., the upper and the lower levels, to address the planning, and operation viewpoints, simultaneously. In the upper level, two multiobjective optimization techniques, including multi-objective non-dominated sorting genetic algorithm (NSGA-II) [24] and multi-objective particle swarm optimization (MOPSO), [25] are employed to maximize two objectives, including HC maximization of the distribution networks and maximization of the total power loss reduction. In the lower level, the graphically based DNR algorithm in Reference [18] takes place to maximize the power loss reduction of the distribution network at each scenario.
In this work, the main contributions are listed as follows: (1). Optimal allocation of WTs and PVs is performed while considering their uncertainties, including wind speed and solar irradiance, respectively. (2). Load uncertainty is considered in this study to step on the real-life benefits of DGs penetration during load alterations. (3). A bilevel multi-objective optimization approach is formulated to optimally size renewable WT/PV DGs along with network optimization from the planning and operational perspectives. Further, the optimal solution is chosen from the pareto solutions via a decision-making algorithm called 'Technique for Order of Preference by Similarity to Ideal Solution' (TOPSIS). (4). Case studies are conducted on real distribution networks, including the 59-node distribution network in Cairo and the 83-node distribution network of the Taiwan power company. Furthermore, the proposed optimization approach is tested on the 415-and 880-node large distribution networks, which are ensembled from the 83node real distribution network.
The organization of this work is enclosed in five sections. Section 2 provides the problem statement of this work, which is composed of the power flow equations, the DNR graphically based algorithm, the WT, and PV modeling, the used scenarios generation algorithm, and TOPSIS. Section 3 formulates the bilevel multi-objective optimization approach used in this paper. Section 4 encloses the results and commentary on the obtained results. Finally, Section 5 briefly illustrates the research outcomes and future recommendations.

Materials and Methods
In this section, the power flow equations, the graphically based DNR mathematical algorithm, load, WT, PV DGs' models, TOPSIS, and system performance indices are illustrated in detail. Figure 1 demonstrates distribution network modeling for the upcoming case studies.

Power Flow Equations
The power flow equations used to calculate injected active/reactive power and the nodal voltages are illustrated as follows [18]: where P k and Q k are the components of the apparent power injected to the kth node; P L k and Q L k are the load's components at the kth node; r b and x b are the impedance components at the bth line; V k is the nodal voltage at the kth node; and B node and B line are the set of nodes and lines, respectively.

Distribution Network Reconfiguration
Recently, in 2019, a novel graphically based DNR mathematical algorithm [18] was proposed by the authors to find a near-global/global positioning of tie-lines for large distribution networks. It has proven its ability to find near-global solutions in a short computational time for large distribution networks up to the 4400-node distribution network. The reconfiguration procedure takes place by exchanging the status of the existing tie-lines of the best configuration (X best rec ) obtained with their neighboring sectionalized lines, after being prioritized by an efficient index called weighted voltage deviation index (WVD) to obtain many temporary configurations (X temp rec ), which are further checked for optimality. Furthermore, this DNR mathematical algorithm has proven its ability to reconfigure the distribution networks without the need for a radiality check, and unlike the other heuristic optimizers, it does not depend on any random generations. The reader can refer to the details of this methodology in Reference [18]. The MATLAB code used to reconfigure the 59-node distribution network is provided in Reference [26], in which the MATPOWER [27,28] toolbox is used for the power flow solution. Figure 2a-c illustrates the status exchange procedure between the tie-lines (dotted lines) and their neighboring sectionalized lines (highlighted in red) to obtain a better loss minimization. In the first iteration, as shown in Figure 3b, the tie line connecting the nodes 18 and 50 is changed to sectionalized (highlighted in red), and the sectionalized line joining nodes 49 and 50 is changed to a tie-line (highlighted in green) to obtain a better configuration that provides a minimum power loss than that in the initial configuration. The convergence curves for power loss minimization using this DNR method are shown in Figure 3 for the 59-, 69-, 135-, and 415-node distribution networks [18], where they reached the optimal fitness at the 11th, 6th, 14th, and 55th iteration, respectively.

DG modeling
Two DG types, WT and PV, are considered in this work. Detailed modeling of these DGs is illustrated in the following subsections in detail.

Wind Turbine DG
WTs are characterized by their intermittent nature due to the variations occurring in the wind speed. To model the WT, three wind speeds are mentioned to characterize the installed WT, including rated speed (v rated ), cut-in speed (v cut−in ), and cut-out speed (v cut−out ). Thus, the WT injected power (P DG−WT u WT ,s ) at the u WT node [29,30] for the sth scenario is expressed as follows: where, v s is the wind speed at the sth scenario, S WT−size u WT is the size of the installed WT at the u WT node, and S rated WT is the maximum capacity of the installed WT.

DG modeling
Two DG types, WT and PV, are considered in this work. Detailed modeling of these DGs is illustrated in the following subsections in detail.

Wind Turbine DG
WTs are characterized by their intermittent nature due to the variations occurring in the wind speed. To model the WT, three wind speeds are mentioned to characterize the installed WT, including rated speed ( ), cut-in speed ( ), and cut-out speed ( ). Thus, the WT injected power ( , ) at the node [29,30] where, is the wind speed at the sth scenario, is the size of the installed WT at the node, and is the maximum capacity of the installed WT.

Solar Photovoltaic DG
PVs are characterized by their intermittent nature due to the variations occurring in the solar irradiance (G). To model the PV DG, two specific irradiance points are characterizing the output power from the PV unit [29,30], including the standard solar irradiance (G std ) and a specific irradiance threshold (R c ). The solar PV penetration is controllable since it can be controlled through power trackability or by charging batteries, thus the actual penetration of a PV DG (P DG−PV u PV ,s ) at the sth scenario is modeled as follows [29,30]: where G s is the solar irradiance at the sth scenario, S PV−size u PV is the size of the installed PV unit at the u PV node, and S max PV is the maximum capacity of the installed PV.

Scenarios Reduction
In this work, the wind speed, the solar irradiance, and the load data are available in Reference [31]. These data are applied to the studied distribution networks to imitate the uncertainty effect on HC maximization. The solar irradiance, wind speed, and load profiles for one year are provided in Figures 4-6, respectively. The 8760 hourly data are reduced to relevant 30 scenarios using the backward reduction technique developed by Growe-Kuska et al. for stochastic programming [32]. The obtained scenarios, including loading level (LL s ), wind speed, solar irradiance, and their probabilities (p s ), are provided in Table 1 for each scenario (s). It is well noted that increasing the number of scenarios will increase the Mathematics 2021, 9, 26 8 of 31 accuracy of the obtained results; however, conducting the optimization procedure based on 30 scenarios is relevant from the practical perspective [30].

TOPSIS
TOPSIS was firstly proposed in the eighth decade of the previous century by Hwang et al. [33]. After that, it has been improved by Yoon [34] in 1987 and by Hwang et al. [35] in 1993. TOPSIS takes place by choosing the preferable alternative (paretosolution) that has the smallest geometric distance from the positive ideal solution and also has the farthest geometric distance from the negative ideal solution. TOPSIS algorithm composed of the following phases: , where m and n denote the number of alternatives and the criteria, respectively. A vector of preset weights ω l is established for each criterion in which the sum of its weights equals one. After that, a matrix (N) called the 'normalized matrix' is established, where N = e ql m×n , and its elements e ql are obtained using the following equation: Phase 2:A new matrix (W N) is calculated, whose dimensions are m × n, and its elements O ql are calculated as follows: Phase 3:At this phase, the best and the worst alternatives are denoted by the 1 × n vectors: E and F, respectively. The elements of E and F are denoted by D l and G l , respectively.
where J − and J + are the negative and positive criteria, respectively. Phase 4:For each alternative, the least-squares distances between the qth alternative and D l and G l are expressed in Equations (12) and (13), respectively. where d best q and d worst q are the distance of each alternative from the best and the worst elements, respectively. Phase 5:At this phase, the similarity index for the qth alternative (SI q ) expressed in Equation (12) is calculated to sort the alternatives.
where SI q belongs to the interval [0,1]. Phase 6:Display the best alternative having the highest SI q value.

System Performance Indices
In this work, two performance indices are used to assess the overall operational performance of the studied distribution networks at different scenarios, including the load balancing index (LBI), the aggregated voltage deviation index (AVDI), and the aggregated fast voltage stability index (FVSI).

Load Balancing Index (LBI)
The LBI was previously used in many publications to assess the loading carried by each line [36]. The LBI was used many times to obtain a better distribution of currents among the lines of the distribution feeders. The formulation of the LBI is illustrated as follows: where N s is the total number of scenarios, and I b,s is the magnitude of the branch current flowing in the bth branch at the sth scenario. I rated is the maximum line current. LBI b,s is the LBI at the sth scenario for the bth line, LBI s is the aggregated LBI for all lines at the sth scenario, and LBI ov is the overall LBI for all scenarios.

Aggregate Voltage Deviation Index (AVDI)
The AVDI was previously employed in many research works like Reference [5] to provide an insight into the voltage deviations from the unity, where the lower AVDI indicates better voltage security at the demand node. The AVDI is formulated as follows: where AVDI s is the aggregated voltage deviation index at the sth scenario, V k,s is the magnitude of the kth node at the sth scenario, and AVDI ov is the overall aggregated voltage deviation index for all scenarios.

Fast Voltage Stability Index (FVSI)
The FVSI was proposed in References [37,38] as a measure for the voltage security at each node. The lower the value of the FVSI, the more voltage security obtained. The fast voltage stability index is formulated as follows: where FVSI b,s is the FVSI of the bth line at the sth scenario, Z b,s is the impedance of the bth line at the sth scenario, whose reactive component is x b,s , FVSI s is the aggregated FVSI for all distribution system lines at the sth scenario, and FVSI ov is the overall aggregated fast voltage stability index for all scenarios.

Problem Formulation
In this section, the objective function and the constraints are illustrated in detail.

Objective Function
In this paper, a bilevel multi-objective optimization problem is formulated for HC maximization while considering network losses. The objective function in this work is twofold, including the upper-and the lower-level optimization approaches. On the one hand, the upper-level optimization problem in Equation (28) is formulated as a multiobjective optimization problem to maximize the probabilistic hosting capacity (PHC) of DGs expressed in Equation (25), and also maximize the total power loss reduction (RP loss ) expressed in Equation (27). On the other hand, the lower-level optimization approach takes place using the DNR mathematical algorithm to choose the appropriate configuration providing a better reduction in the power loss; thus, the objective function at this level is Equation (25). The pseudo-code for the proposed stochastic optimization approach is provided in Algorithm 1.
RP loss (%) = 100 P 0 loss − P loss P 0 where HC s is the HC at the sth scenario, P loss is the probabilistic total active loss for all the studied scenarios, and P 0 loss is the total power loss at the normal loading conditions.

Constraints
In addition to the constraints of WT, PV DGs rated capacities expressed in Equations (5) and (7) and the FVSI constraint provided in Equation (23). The following operational/planning constraints are applied in the optimization process.
where P slack s in the active power delivered by the substation at the sth scenario, and V min and V max are the lower and upper nodal voltage limits, respectively.

Algorithm 1
The proposed bilevel multi-objective optimization for HC maximization 1.
Set the number of populations, number of iterations, number of decision variables, and variables' limits.

2.
While iteration number smaller than or equal to the total number of iterations 3.
Update the upper level multi-objective optimization parameters, including S WT−size u WT and S PV−size u PV .

4.
Input the available WT and PV nodes.

5.
Set s equal to one. 6.
While s ≤ N s 7.
Apply LL s to the connected loads.  (27). 11. The lower level optimization problem takes place at this sub-step by reconfiguring the existing tie-lines using the DNR mathematical algorithm. 12. If Equations (5), (7), (23), and (29)-(32) violated 13. Then set the PHC and RP loss value to zero, increment the iteration number, and return to Step 2. 14. End While 15. Evaluate PHC and RP loss expressed in Equations (25) and (27)

Results and Discussion
In this work, five distribution networks, 16-, 59-, 69, 83-, and 415-node distribution networks are used [39][40][41][42]. The 415-node distribution network is a large distribution network, composed of five instances of the 83-node real distribution network to mimic the complexity of large real distribution networks. The schematic diagrams of the 59-and 83-node distribution networks are provided in Figures 7 and 8. The input data used for the upcoming case study is supplied in Tables 2 and 3. In this work, two multi-objective optimization approaches are employed to solve the upper multi-objective nonlinear optimization problem, including non-dominated sorting genetic algorithm (NSGA-II) [24] and the multi-objective particle swarm optimization (MOPSO) [25]. NSGA-II is one of the most well-known multi-objective optimizers. It was developed by Deb et al. in 2002 as an improvement for the genetic algorithm to handle multi-objective optimization problems via several characteristics in its code, including fast non-dominated sorting and crowded distance estimation. The reader is referred to Reference [24] for more details about this multi-objective optimization algorithm. In the particle swarm optimization algorithm, each solution is known by its position and velocity, where the best solution of the particles at a certain iteration is denoted by pbest, and the global solution obtained till the current iteration is denoted by gbest. A further extension was conducted in Reference [25] to handle multi-objective optimization problems. The reader is referred to Reference [25] for more details about MOPSO. The flowcharts of the NSGA-II and MOPSO are shown in Figures 9 and 10. Furthermore, the DNR mathematical algorithm in Reference [18] used to reconfigure the studied distribution networks in the lower-level optimization problem.
The following case study is conducted on MATLAB r2018a on a DELL Laptop, its model name is 'Latitude E7450', including an 'Intel ® Core™ i5 CPU at 2.3 GHz, and 8 GB RAM.        Figure 10. Flowchart of the multi-objective particle swarm optimization (MOPSO) algorithm. Figure 10. Flowchart of the multi-objective particle swarm optimization (MOPSO) algorithm. In this work, the WT and PV DGs are allocated at a certain set of nodes due to land space limitations. These PV/WT nodes are provided in Table 4. The PHC is assessed for the 16-, 59-, 69-, 83-, and 415-node distribution networks at different scenarios of wind speeds, solar irradiance, and load uncertainties, as shown in Table 5. The pareto-front of the obtained results for the 83-and 415-node distribution networks are provided in Figures 11 and 12, respectively. The WT and PV DGs sizes in MW at each node using NSGA-II and MOPSO, are provided in Tables 6 and 7, respectively. Besides, the configurations of the 16-, 59-, 69-, 83, and 415-node distribution networks at each scenario are provided in Tables 8-12, respectively. The HC and power loss reduction are provided in Figures 13 and 14, respectively, for the 59-, 83-, and 415-node distribution networks at each scenario. Finally, the voltage profiles at multiple scenarios for the 59-and 83-node distribution networks are provided in Figures 15 and 16, respectively.   Figure 11. Pareto-front for the 83-node distribution network using NSGA-II and MOPSO.

HC (%)
Dominated solutions Pareto optimal solutions (NSGA-II) Pareto optimal solutions (MOPSO) Figure 11. Pareto-front for the 83-node distribution network using NSGA-II and MOPSO.                 As a result of changing the topological structure of the studied distribution networks shown in Tables 8-12, the capability of accommodating more WT/PV DGs has increased while considering solar irradiance, wind speed, and load uncertainties. The achieved benefits from employing DNR in each scenario are illustrated as follows from the planning and operational perspectives.
From the planning perspective, the HC using the NSGA-II has reached 17.7173%, while in the case of MOPSO, the HC has reached 17.5299% for the 415-node distribution network. The power loss reduction reached for NSGA-II and MOPSO is 74.5320% and 79.2044% for the 880-node distribution network, respectively. Thus, it is hard to assume that an optimization technique is better than the others for the proposed optimization problem. Moreover, the allocated WT/PV DGs are not suitable for all the available nodes of the distribution networks, as in the 415-node distribution network, the number of available nodes for WT, and PV DGs is 50 for each of them; however, the number of allocated WT and PV DGs is 19 and 20, respectively, using NSGA-II. Besides, at scenarios 1, 11, 17, and 26, the HC was zero as the solar irradiance was zero, and the wind speed was zero below its cut-in speed, thus the generated WT and PV powers were zero at these scenarios. To overcome this issue, energy storage systems (ESSs) should be optimally allocated along with WT/PV DGs allocation as a solution to overcome the problem mentioned above at times of lower irradiance and wind speeds. ESSs allocation is beyond the scope of this work and will be mentioned in upcoming future work.
From the operational perspective, three operational indices were employed to measure the improvement in the system's load balancing and voltage security, including LBI ov , AVDI ov , and FVSI ov . The results obtained for these indices using NSGA-II and MOPSO are different for each distribution network, as demonstrated in Table 5. Thus, it is difficult to conclude that one of them is better than the other. From the LBI ov viewpoint, the LBI ov decreased by 30.0472%, 34.4887%, 51.7110%, 29.6219%, 11.7869%, and 69.1087% for the 16-, 59-, 69-, 83-, 415-, and 880-node distribution networks using NSGA-II. From the AVDI ov viewpoint, the AVDI ov decreased by 32.4932%, 40.0142%, 62.9626%, 19.8607%, 9.6610%, and 70.2924% for the 16-, 59-, 69-, 83-, 415-, and 880-node distribution networks using NSGA-II. From the FVSI ov viewpoint, the FVSI ov decreased by 10.0186%, 17.5676%, 34.8239%, 4.4225%, 0.9656%, and 39.7736% for the 16-, 59-, 69-, 83-, 415-, and 880-node distribution networks using NSGA-II. Moreover, the voltage profile improvement shown in Figures 13 and 14 validates the effectiveness of the proposed optimization approach independent of the used optimization technique. Finally, a multiple-scenario analysis was conducted on the 59-node distribution system to ensure the effectiveness of choosing 30 scenarios for the previous case studies. Table 13 gives insight into the effect of changing the number of generated scenarios on the obtained HC and power loss reduction for the 59-node distribution network in Cairo. From Table 13, it is notable that the obtained HC ranges between 17% and 19%, its average is 17.9153 %, and its standard deviation is 0.8336 using NSGA-II. Thus, choosing 30 scenarios is relevant for our studied cases. Moreover, to ensure the effectiveness of choosing NSGA-II and MOPSO against the other multi-objective optimizers, a comparison with multi-objective multi-verse optimization (MOMVO) [42] and multi-objective flower pollination algorithm (MOFPA) [43] was conducted to step on their effectiveness. As shown in Table 14, the NSGA-II and MOPSO have provided better near-optimal solutions; however, they took a long time in computation than MOMVO and MOFPA. Besides, we aim to maximize the HC of the studied distribution systems while improving loss reduction. In this regard, NSGA-II and MOPSO were chosen to solve the optimization problem since they provided the best HC and power loss reduction. Finally, a comparison with previous works based on system's performance indices for the 83-, 415-, and 880-node distribution networks is shown in Tables 15-17, respectively, to clarify the effectiveness of the proposed optimization algorithm from the operational perspective.

Conclusions and Future Works
In this paper, a novel optimization approach was deployed in the form of multiobjective bilevel optimization for maximizing both the HC and the losses reduction percentages using a graphically based DNR mathematical approach. Five distribution networks were tested for accommodating WT and PV while considering solar, wind, and load uncertainties. In comparison with the previous works [15][16][17]41,45] used to improve the operational indices of the studied distribution systems, the proposed optimization approach via allocating WT/PV succeeded in minimizing the total active loss of the 83-, 415-, and 880-node distribution networks by 80.7985%, 74.5320%, and 93.5010%, respectively. Furthermore, the voltage profiles shown in Figures 15 and 16 reinforce the obtained outcomes from using DNR along with WT/PVs allocation and ensures the ability of the proposed strategy to keep the voltage within its permissible limits. The proposed optimization approach succeeded in maximizing the HC, and the power loss reduction for the studied distribution networks by greater than 17% and 74%, respectively. The advantages of this work were the effectiveness of applying DNR to maximize the HC of real/large distribution networks while considering DGs uncertainties and also improving the system's operational indices and enhancing loss reduction. Other points that were not under the scope of this work and will be addressed in future works are the reliability evaluation of the obtained configurations via system average interruption frequency index (SAIFI), system average interruption duration index (SAIDI), and other reliability indices and considering other power quality issues that affect the HC of the distribution systems. Future works will consider the allocation of multi-terminal SOPs [46] and various energy storage types [47] along with DNR for increasing the HC of the studied distribution systems while ensuring an effective protective scheme [48]. Besides, reliability will be considered in the possible works [49][50][51] to provide a reliable configuration for unbalanced large/real distribution networks. Furthermore, power quality enhancement strategies will be considered while considering compliance with the IEEE standards [52].

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to its large size.

Conflicts of Interest:
The authors declare no conflict of interest. Overall LBI for all scenarios N s Total number of scenarios PHC Probabilistic hosting capacity P k + Q k i Apparent power injected to the kth node P L k + Q L k i Load's apparent power connected to the kth node p s Probability of the sth scenario P loss Probabilistic total active loss for all the studied scenarios P 0 loss Total power loss at the normal loading conditions P slack s Active power delivered by the substation at the sth scenario R c Specific irradiance threshold r b + x b i Impedance of the bth line RP loss Total power loss reduction S max PV Maximum capacity of the installed PV S rated WT Maximum capacity of the installed WT V k Nodal voltage at the kth node V k,s

Nomenclature
Magnitude of the kth node at the sth scenario V min Lower nodal voltage limit V max Upper nodal voltage limit v rated WT rated speed v cut−in WT cut-in speed v cut−out WT cut-out speed v s Wind speed at the sth scenario