Enhancing the Resilience of Hybrid Microgrids by Minimizing the Maximum Relative Regret in Two-Stage Stochastic Scheduling

Occurrence of severe events with high-impact and low-probability (HILP) can cause significant disturbances to power grids. The strength of the system to withstand such HILP events is interpreted as system resilience. This paper proposes a novel approach for hybrid AC/DC microgrids with the aim of resilience enhancement and based on a linear stochastic two-stage scenario-based minimax relative regret (LSTS-MMRR) optimization according to the optimality robustness concept. Photovoltaic arrays and wind turbines, microturbines, and energy storage systems are scheduled based on the proposed approach in microgrids. Considering the uncertainty of emergency duration due to disruption from the upstream grid, the optimization problem is decomposed into standard and critical stochastic situations. This work also quantifies the importance of uncertainty with the well-known quantities named Value of Stochastic Solution (VSS) and Expected Value of Perfect Information (EVPI). For both normal and emergency circumstances, exclusive objective function and constraints are considered. Meanwhile, time-related variables maintain their continuity between normal and emergency circumstances. Therefore, the proposed approach minimized the maximum relative regret of objective function through defined scenarios. The proposed technique is compared with two common methods in resilience enhancement issues, and its effectiveness is demonstrated through an analysis of the VSS and EVPI.

Amount of objective function in emergency circumstances at each scenario.

I. INTRODUCTION
Enduring low-probability (LP) and high impact (HP) severe events without experiencing any major disturbance according to the system ability is interpreted as resilience in power systems [1]- [3]. Due to the growing repetition of the severe events related to weather (e.g., storms, earthquakes, and floods), the resilience issue becomes particularly important [3]. In this regard, microgrids (MGs) are employed in the power systems' resilience enhancement due to their various energy sources, including energy storage systems (ESS), renewable energy sources (RES), and fuel distributed generations (DG) [4], [5]. In order to increase resilience, several criteria for assessing resistance have been proposed in the literature. Load shedding/ load curtailment minimization or system curtailed load penalty cost minimization has been recognized as an optimization-based evaluation criterion of system resilience in [6]- [9] and [10]. Another resilience evaluation criterion applied by authors is the recovery rate. Restoration of vital load maximization after the severe event has been considered in [11], [12], and [13]. Authors in [14] considered energy supplied to critical loads maximization as a resilience evaluation criterion. Expected energy not supplied minimization and expected load curtailment minimization have been approved as resilience evaluation criteria in [15] and [16], respectively. Load shedding minimization according to loads priority is the aim of [17]. In the mentioned literature [4]- [17], resilience assessment and improvement methods are specified through optimization problems. Therefore, resilience assessment and improvement techniques of power systems require suitable optimization problem determination [3]. On the other hand, due to the inherent uncertainties in a power system, such as uncertainties related to renewable energy generation, loads, and emergency duration in critical situations due to severe event occurrence, it is essential to present appropriate stochastic models to deal with such issues. Uncertainty optimization approaches applied in studies are generally classified into two categories. A risk-neutral scenario-based method is employed in some following investigations. In [16], a scenario-based stochastic mixed-integer linear programming (SSMILP) model, according to a risk-neutral approach, is proposed to enhance the resilience of power systems. By applying a two-stage stochastic framework based on riskneutral programming, the renewable energy resources penetration impact according to their related uncertainties on system resiliency is investigated in [11]. A two-stage stochastic post-hurricane framework with load recovery purpose is suggested in [14] to improve networked MG resilience via reconfiguration strategy in the first stage and mobile emergency resources application in the second stage. In [6], for the MGs' load shedding decrease in severe events, a risk-neutral scenario-based stochastic programming model is proposed. In the second category, based on a risk-averse approach, a robust model is used in some research. For example, authors in [7] have formulated an optimization model based on a two-stage robust day-ahead scheduling with the aim of minimizing MG operation cost while the load curtailment and frequency deviation can be controlled through events that caused islanding. In [18], a robust dispatch on islanded hybrid AC/DC MGs has been conducted emphatically, and uncertainty of load and power sources have been considered. A robust mixed-integer second-order cone programming (R-MISOCP) model is developed in [9] for islanded MGs due to main grid interruption, based on the MGs resilienceoriented optimal scheduling. In [17], to manage MGs against natural disasters, a proactive conservative schedule framework is proposed. Furthermore, a stochastic-robust optimization method is introduced in [10] to schedule MG in an optimal manner under the standard and emergency operation mode.
According to John M. Mulvey's interpretation [19], to cope with optimization problems under uncertainty, the proposed models should guarantee Feasibility Robustness (FR) and Optimality Robustness (OR). OR guarantee confirms that the mathematical programming optimal solution will be robust concerning optimality if it remains ''close'' to optimal for any realization of the possible scenarios. According to [20], robust solutions to optimization problems are those that enjoy both feasibility and optimality robustness at the same time. In terms of feasibility robustness, an optimization solution remains feasible for all (almost) possible values of contingent input parameters, and when the solution complies with optimality robustness, for (almost) all possible values of uncertain input parameters, the objective function for the solution stays near the optimal value or has a minimum (undesirable) deviation from the optimal value. Consequently, in [4]- [17], OR guarantee is not considered. The minimum deviation of the optimal solution in each scenario is an important feature in programming under uncertain conditions that results in minimum regret in each scenario. Therefore, in this paper, a two-stage stochastic scenario-based (TSS) approach based on the Minimax Regret model (MMR) is proposed, with the aim of hybrid MGs resilience enhancement. Since the emergency duration follows a specific probability distribution function, the normal and emergency operating conditions are modeled under a stochastic approach. Therefore, the objective function and constraints are considered differently in normal and critical conditions in this paper. Minimizing the operation cost of hybrid MGs considering the power exchange between upstream grid according to power market is considered as normal operation objective function. In addition, minimization penalty cost of load shedding according to load priority in AC and DC MGs leads to minimizing load curtailment as a resilience objective function in an emergency condition. Optimal scheduling of energy storage, micro-turbines, and renewable energies according to their technical constraints are obtained in this study in all operating conditions. The novel proposed method in this study is such that according to different scenarios, the least regret occurs based on the emergency duration in stochastic objective functions. Table 1 he comparison of the principal highlights of the previous investigations and this paper. The notable contributions of the proposed method are declared as follows: • Developing a novel linear stochastic two-stage minimax relative regret (LSTS-MMRR) based on resilience enhancement of hybrid AC/DC MGs.
• Sensitivity analysis at the disruption start time in a resilience framework guaranteeing optimality robustness • Maintaining reciprocity between emergency and normal situations in hybrid MGs functions.

II. PROPOSED LSTS-MMRR METHODOLOGY
In this paper, a new approach is proposed concerning resilience enhancement in hybrid AC/DC MGs. According to uncertain emergency condition duration, linear two-stage stochastic scenario-based programming is modeled. Further, it is assumed that emergency durations follow a normal distribution with a mean of 5 hours and a standard deviation of one hour [22]. Therefore, there are different situations for scheduling the sources and various manners to manage the system. Hence, the proposed method is a new approach to obtain the optimal answer so that minimum relative regret is achieved in all scenarios. The regret value is defined as an objective function, which contains all scenarios. For example, in a real decision taken by a system operator, a decision is considered an appropriate one that has the minimum deviation from all scenarios.
where x and y s are the H&N and W&S decision variables, respectively. In addition, δ T , ρ T s , λ s , ϑ s , α and β are known coefficients. If the optimal solution is obtained individually under each uncertainty scenario, it will be (2): For a given S, ξ * s and x * , y * s define the objective function and the decision variables in optimal values of the above problem, respectively. If x is selected as the decision variables, then the relative regret associated with having decided x rather than x * can be defined as follows (3): [23] Relative Equation (3) is considered for all scenarios. Then to find out the maximum deviation of the optimal solution in each scenario, equation (4) is used: [23] After determining the maximum deviation of the optimal solution of the defined scenario, we aim to minimize the maximum relative regret according to Eq (5) Actually, the maximum relative regret among all possible scenarios has been minimized, and it can be guaranteed that the optimality robustness. In fact, the maximum distance from optimality is minimized in this issue, and this is a risk-averse approach. The proposed new method in resilience enhancement framework and its performance in this paper on hybrid AC-DC MGs is presented in Fig.1.

III. MATHEMATICAL FORMULATION
In order to increase the hybrid MGs resilience, the mathematical formulation based on the proposed method is presented here. As mentioned in the Introduction, due to various operation conditions, objective functions and technical constraints are considered appropriate to each situation. In the following, the goals and constraints in normal and emergency conditions are illustrated: In normal conditions, an economic point of view in MGs' cost function is investigated. The objective function is formulated as (6): The first term of the cost function represents the MTs' operation cost, start-up cost, and shut-down cost in the both MGs, respectively.
Eq. (6.2) corresponds to the operation cost of wind and solar energy sources, sequentially. The operation cost of ESSs is matching to Eq. (6.3). Eq. (6.4) is related to standard operating situations and depict the price of buying power from the upstream network and the profit from the sale of power to the upstream network. Furthermore, Eq. (7) illustrates the proposed method formulation as follows: In Eq.7 the optimal solution in each scenario should be attained. To be more specific, the optimal objective value is obtained according to the optimal value of variables (Eq.7.1) Power balance constraints for DC and AC MGs, which are shown in Eqs. (8) and (9), affirm that at each time scheduling and for each scenario, the total power generated by MTs and RESs, ESS charge/discharge power, and exchanged power The limitations of buying and selling power exchanged from point of common coupling (PCC) of utility grid are indicated in Eqs. (10) and (11). To prevent buying and selling simultaneously, Eq. (12) is employed.
Satisfying the interlinking converter (ILC) constraint is implemented in Eq. (13) at DC MG as follows: Total bought and sold power according to each MG's trading power and the its limitation is shown in Eqs (14)- (16): Load shedding of MGs isn't permitted in normal operation according to (17): Furthermore, the stochastic unit commitment (SUC) problem is implemented in MGs all over the situations based on [24]. Constraints related to ramp rate, minimum up/downtime, and on/off status of MTs are shown in Eqs(18)- (26) Note that in these equations, m∀acm, dcm is considered.
Y t,m,g ≤ X t+MT (m,g),m,g (25) Z t,m,g + X t+MT (m,g),m,g ≤ 1 (26) Given the presence of RESs and the advantages of ESSs as well as their effects on resilience, formulation of ESSs is be presented in MGs [22] as exhibited in (27)-(31): The equation related to state-of-charge (SOC) and charging/ discharging power is shown in (27) according to battery efficiency. Permissible SOC limits due to battery lifetime constraint are presented in Eq. (28). The limitation of maximum charging/discharging power, as well as the simultaneous charging and discharging prevention, are shown in Eqs. (29)-(30) and (31), respectively. Note that in these equations, m∀acm, dcm is considered.

B. STOCHASTIC EMERGENCY OPERATION 1) OBJECTIVE FUNCTION
Immediately after the disaster occurs, the system enters an emergency state. After switching the system status, the variable values before the event are considered as an initial state for the emergency status variables. Meanwhile, the purpose of the problem also changes based on the resilience enhancement of MGs. Minimizing the load shedding according to their priority is in the first preference of the objective function in an emergency state. A new term as load curtailment of each MG is considered (of 5 in (32.1)) in (32), according to their priority importance. Furthermore, Eq. (33) illustrates the proposed method formulation in the emergency condition.
According to Eq.33, the optimal solution should be reached in every scenario. More specifically, the optimal objective value is determined by the optimal value of the variables (Eq.33.1).
It is noteworthy that in addition to the above limitations, equations (27)-(31) must also be considered in an emergency condition due to the interlink and coordination of two states of the system.

C. RESILIENCE METRIC CALCULATION
For evaluating the MG resilience in different disaster scenarios, resilience metrics are utilized. Resilience metrics is a proper means for analyzing the system resilience in different conditions while severe circumstances prevail in the system. To measure MGs resilience, Eq. (40) is used according to the load provided in each MG at the emergency condition and the total load available at that time [25]. MGs load priority is considered in this formulation by Eq.(41) according to the highest and the lowest importance [21].
Lsh ω t,m,ρ Dm t,m,ρ × pri ρ N RM max (40) In this work, the objective is to minimize the maximum relative regret, which in the context of resilience refers to the economic loss minimum caused by an extreme event.
As a result, load shedding is minimized in emergency situations. Thus, minimizing load shedding can be seen as an evaluation criterion that is based on minimizing maximum relative regret, which is a propped strategy for enhancing resilience. In the proposed method, resilience evaluation criteria, resilience enhancement strategy, and resilience metric are all linked to the amount of load shedding provided by the optimization problem.

D. EVPI AND VSS MODEL
The effectiveness of the proposed method can be evaluated by expected value of perfect information (EVPI) and value of the stochastic solution (VSS). A decision-maker can calculate the maximum amount they will be willing to pay if all information is perfectly known. To put it in other words, if the uncertainty can be eliminated, a risk of receiving unexpected outcomes can be reduced to zero [26], [27]. EVPI quantifies this effect, which describes how having complete information about uncertainty is beneficial. By summing the optimal solutions for each scenario multiplied by the probabilities they have, the wait-and-see solution (WS) is calculated. If PS is considered the solution of the proposed method, we will have Eqs.(42) and (43) as follow: VSS is a product of explicitly modeling uncertain distributions. Mathematically, it is defined as the difference between the expected value (EV) where uncertain variables are replaced by their mean values, and the stochastic solutions [27]. VSS value is formulated in Eq.(44) The comparison of first-stage decisions in an uncertain circumstance and decisions based on the assumption that parameters take their expected (mean) values is made by VSS to determine whether the inclusion of uncertainty benefits the decision.

A. TEST SYSTEM UNDER STUDY
The model was coded in GAMS environment and solved by the commercial solver GUROBI analytically. The proposed model is implemented in hybrid AC/DC MGs as is exhibited in Figure 2, to assess the performance and efficacy of the suggested approach. In AC MG, three micro-turbines as common DG, a wind turbine as renewable DG unit, decomposed loads into three levels (priorities), and ESS units are considered. In DC MG, two micro-turbines as common DG, a photovoltaic array as renewable DG unit, decomposed loads into two levels (priorities), and ESS units are considered across the 24-hour time horizon [21]. In addition, time-of-use market price signals are borrowed from [21]. MGs' prioritized loads and RES generation in 24 hours are represented in Fig. 3 and Fig. 4. The parameters of the test system are given in Table 2 and Table 3.  Further, emergency durations are considered as some scenarios with certain occurrence probability in Table 4. The durations of unscheduled emergency circumstances should be analyzed to determine the probability distribution function (PDF) that represents the associated randomness the best. In order to use this PDF in our stochastic framework, it is approximated as several discrete scenarios [28]. It is assumed that emergency durations follow a normal distribution with a mean of 5 hours and a standard deviation of one hour [22].   In this regard, three case studies are considered as follows: • Case I : expected value optimization method. • Case II : proposed optimization method. • Case III : worst-case optimization method.

B. SENSITIVITY ANALYSIS ON EMERGENCY START TIME
Since the severe event start time is unknown, a comprehensive sensitivity analysis is investigated. Therefore, the emergency condition start time is considered in all 24 hours of the time horizon scheduling to determine the worst start time based on the objective function's value. Fig 5 depicts the expected and worst cost through all possible disruption start times in the proposed method. The worst disruption start time according to operation cost, which contains load shedding cost, is at 1:00. In other words, if a disaster occurs at 1:00, the worst condition will emerge.
The difference in expected operation cost of the proposed method and the first case is insignificant compared with the amount of difference between the third method and the proposed method which is illustrated in Fig 6. The efficiency of the proposed method is obvious compared to the third method, especially in some emergency start times like 17:00-24:00. For example, if the disaster occurs at 24:00 and 23:00 and lasts an average of 5 hours, the expected operation VOLUME 10, 2022  cost of the proposed method is approximately 7290$ less than the case III.
The values for the worst-case operation costs are also displayed in Fig.7. It is noticeable that the first case approach reports higher amounts of operation cost if considering the worst-case state. This is while the amount of operation costs in the proposed method is not much different from the third case. For example, if the disaster occurs at 1:00 and lasts an average of 5 hours, the worse-case operation cost of the proposed method is approximately 4600$ less than case I and only 1800$ more than case III. From Figs. 6 and 7, it can be concluded that the values related to the expected and worst operation costs in the proposed method compared to the same values as the other two common methods have the positive features of both methods. On the other hand, to evaluate the results of sensitivity analysis, Fig. 8 is used, which is the interpretation result of Figs. 6 and 7. The ratio of the total load to be supplied and total available RES during the critical situation to the total load and available RES of the MGs, respectively in a studied period is shown in Fig 8. It can be asserted that only 19.8% and 3% of available WT and PV total generation during the emergency time started at 1:00 can be operated. However, the maximum wind and solar available energy in MGs are 46% and 89% respectively when an emergency condition occurred at 9:00. Moreover, the total experienced load in the critical time to a total load of all periods' ratio is equal to 31% and 28% in AC and DC MGs when disaster starts at 1:00. Meanwhile, the required load to supply in all emergency star time deviates up to 14% and 5% in DC and AC MGs, respectively. Therefore, unlike considerable changes in RES generation, the amount of load supplied in critical times does not alter significantly. As can be observed, during the hours when PV generation is significant, load curtailment during the emergency condition and consequently the operating cost of MGs is less than the other emergency circumstances. This interpretation also applies to WT generation. The results of this section conclude that the amount of load required to be provided and also RES generation are effective factors in identifying the worst time of disaster occurrence according to defined resilience evaluation criteria. Therefore, according to Fig. 5, the worst time to start an emergency is at 1:00.

C. PROPOSED NOVEL LSTS-MMRR METHOD RESULTS
In this section, a comprehensive analysis is performed on the relative regret values per hour from the disaster start time, which includes 24 separate scenarios. In each of these 24 scenarios, which are related to the emergency start time, seven other scenarios determine the uncertain duration of the emergency situation. According to section III, it is demonstrated that the optimal answers obtained in each scenario have optimality robustness so that they have the least difference of optimal value in all scenarios. Therefore, the amount of relative regret obtained in each event scenario is displayed in Fig.9. It can be concluded that the regret amounts obtained in each situation depend on the start time of the severe condition although it is guaranteed that the least amount of regret is obtained in each scenario.
For instance, according to Fig.9, the maximum relative regret is 0.18% if a disruption occurs at 1:00. This is while if a disaster starts at 18:00, the relative regret has an amount of just 0.09%. Consequently, depending on disruption start time, the amount of relative regret is various. Table 5 is presented to prove the OR feature of the optimal answers obtained using the proposed approach compared to the other two methods in some random scenarios.

D. ANALYSIS OF THE WORST EMERGENCY START TIME
It was confirmed that the worst disaster start time, according to the sensitivity analysis, occurs at 1:00. Therefore, we implement the proposed method in the worst case to show the efficiency of the suggested approach.
Two main factors to investigate the effectiveness of the proposed methods in resilience problems are the operation costs and load shedding amounts [3]. In this regard, Figs.10 and 11 are depicted which focus on the amount of cost and load shedding in each scenario respectively. According to the different scenarios, the figures compare the amount of cost and load shedding in all cases. If the nominal values of objective functions have been considered, it can be concluded from Fig.10 that the operation cost at all scenarios in Case III has constant values. It means that in scenarios in which the duration of disruption is short, a high operation cost is imposed on the system, which is not desirable. This issue indicates the disadvantage of third case compared with proposed method. In contrast, the quantities related to Case I have reasonable relation with the duration of the disruption. By increasing the duration of emergency circumstance, the nominal operation cost is increasing in Case I. Although the operation cost of Case II has higher amounts compared to Case I in scenarios s1-s5, it has few differences which is negligible. Moreover, the objective function values in Case II at scenarios s6 and VOLUME 10, 2022  s7 have smaller quantities compared with Case I, which is desirable for more longer time of emergency duration.
To further investigate, the total amount of MGs' load shedding in each scenario is shown in Fig 11, applying three methods.
Each scenario shows a different duration of emergency circumstance, so we want to assess the results related to each method. Fig.11 shows load shedding for all scenarios related to each case. The objective function of the first case concentrates on the nominal values, so the load shedding in s6 and s7 has increased because these scenarios have the longest emergency duration. However, the load shedding in s1-s5 has lower quantities. In contrast, in case III the objective function focuses on the worst-case; therefore, scenarios in which the emergency duration is longer have a lower amount of load shedding because they are considered as the worstcase circumstance. By comparing case I and case III, it is concluded that these methods focus on some scenarios due to their aims. Despite the drawbacks of the existing methods, the proposed approach has handled them effectively. In s1-s5 the load shedding has its lowest amount compared to case III. Moreover, in s6 and s7, a lower amount of load shedding can be observed compared to case I. To put it in other words, the load shedding has decreased in case II compared to case I by 70.6% and 68.3% in scenarios s6 and s7, respectively, while it has increased just 11.4% and 15.7% compared to case III at the same scenarios. The average of total load shedding in different scenarios in the proposed method is 15.3% and 3.2% less than case I and case III, respectively. Overall, the average of total load shedding, in different scenarios, in the proposed method is 15.3% and 3.2% less than case I and case III, respectively. In conclusion, the proposed approach takes into account all scenarios in order to minimize load shedding.
For more investigation, the total load shedding within different scenarios according to three methods is shown in Fig 12. For more explanations, in Fig 12.a, the average load shedding in the proposed method is 62.7% and 82.24% less that case I and case III values. Moreover, for Figure 12.b, the average load shedding in the proposed method is 48.5% and 59.58% less that case I and case III values, respectively.
On the other hand, the results are also analyzed from the resilience metric point of view. The resilience metrics difference in the proposed approach compared to other methods is depicted in Fig.13 for AC MG and Fig. 14 for DC MG, while emergency start time is at 1:00 as a worse start time of critical condition. Fig.13 demonstrates that the resilience metric in the proposed method is improved in some scenarios than in the other two methods. It is notable that, for every 5% change in load shedding, the resilience metric changes by 0.0125. For example, in scenarios 6 and 7 at 7:00, the amount of resilience metric in Case II is 26.6% more than in Case I. Furthermore, at 3:00,4:00,5:00, and 6:00 in scenarios 2 to 5, respectively, the resilience metric has increased significantly in Case II compared to Case III. Fig.14 depicts resilience metric improvement in other scenarios and different hours by distinct colors. According to Fig.14, it can be observed that the resilience metric in DC MG in all scenarios is at its maximum value in the proposed method. This indicates that the DC MG doesn't experience any load shedding based on the proposed approach.

E. CALCULATED VSS AND EVPI
Here the values of EVPI and VSS in disturbance start times are illustrated in Fig.15 and Fig.16, respectively. The EVPI, which indicates the importance of addressing the uncertainty of information, indicates how effective the lack of information is. The small amount of the EVPI indicates that obtaining information about the future will not cause a significant impact on the outcomes. In addition, its large amounts can be attributed to the fact that ignoring the importance of completing information is costly. As a matter of fact, EVPI shows how much it can cost to obtain complete information. For example, in Fig. 15, if a disruption occurs at 17:00, the most considerable amount of EVPI can be observed compared with other disruption start times. This amount of EVPI indicates that ignoring the uncertain duration of an emergency at 17:00 is costlier than other times.
The VSS shows how much can be saved by using a stochastic scheduling solution instead of a deterministic scheduling     VSS can be observed compared with other disruption start times. The deterministic approach at 18:00 is clearly not an appropriate solution in comparison to other disruption start times.
Based on the analysis of all disruption start times, the calculated VSS and EVPI are opposites of zero, demonstrating the effectiveness of the stochastic method.

V. CONCLUSION
Given the increasing catastrophic events frequency, how to deal with emergency conditions is an important issue in the power systems operation. MGs as small power systems with distributed generation resources such as renewable generation is considered in the resilience enhancement strategy. Due to the existence of different types of MGs, including DC and AC, in this paper, the coordination and cooperation of hybrid MGs in emergency situations were considered. In this regard, in the paper, a novel resilience enhancement approach for hybrid AC/DC microgrid based on a linear stochastic twostage scenario-based minimax relative regret (LSTS-MMRR) optimization is proposed. According to the uncertainty of event start time, a comprehensive sensitivity analysis is performed on the study time horizon within 24 autonomous scenarios. On the other hand, another uncertainty regarding the critical state duration in the form of a probability distribution function within seven scenarios is considered. Therefore, the objective function is considered by focusing on optimizing economic operation in normal situations and minimizing the prioritized load shedding in emergency situations. It should be noted that there is an economic perspective in critical operation, but it is not the priority of the objective function. The proposed approach is compared with two common methods used in literature as three case studies. In this study, it is concluded that the proposed method, in addition to possessing FR and OR attributes, also has the positive aspects of the other two cases. By implementing the LSTS-MMRR method and comprehensive analysis on disruption start time, it was concluded that the worst emergency condition start time is at 1:00 based on expected and worst-case amount of objective function. Then to demonstrate the efficiency of the proposed method, the results of this method have been compared with two prevalent optimization methods used in previous studies. Decreasing objective function in all condition with a focus on load shedding term was the main goal in this study. In this regard by analyzing all possible disruption start times, it was observed that in 87.5% of disruption start times the expected value of the objective function in case II is less than Case III. Furthermore, it was concluded that in 62.5% of disruption start times the worst-case value of the objective function in case II is less than Case I. This was while the optimality of the outputs was robust. In the following, the obtained relative regret values were presented according to the start time of the emergency operation. For instance, if a contingency occurs at 17:00, the maximum relative regret value in the proposed method is 0.13%, while this value in Case I and Case III is 0.47% and 0.88%, consequently. As a result, in the proposed approach, it is proved that the maximum relative difference of optimality in each scenario has its lowest value in each disruption start time of severe condition. When all disruption start times are considered, the calculated VSS and EVPI take considerable value, which indicates the effectiveness of the stochastic approach.