A Comparative Study Between Bayesian Network and Hybrid Statistical Predictive Models for Proactive Power System Network Resilience Enhancement Operational Planning

Enhancing the operational resilience of the distribution system network (DSN) proactively in a hurricane-prone region requires a pre-hurricane event DSN optimization model, built on accurate hurricane-induced DSN line damage prediction scenarios. In the past, the resilience evaluation methods such as statistical sequential and non-sequential Monte Carlo simulation (MCS) contingency-based technique, and Machine learning-based Bayesian Networks (BN) technique, have been proposed to strengthen the operational resilience of the DSN proactively against forecasted oncoming hurricane events. However, a comparative study is largely unexplored to evaluate which of these two methods is best for proactive operational planning decision-making against forecasted oncoming hurricane events. In this paper, the Bayesian network (BN) and hybrid statistical DSN’s Fragility-curve (FC)-Monte Carlo simulation (MCS)-Scenario reduction (SCENRED) predictive algorithms were developed. The DSN line fault prediction scenarios simulated leveraging the predicted oncoming hurricane Ewiniar data were utilized to perform pre-hurricane DSN optimization to proactively decrease the DSN expected load loss. The pre-event system optimization problems were formulated in a mixed integer linear programming (MILP) approach and solved using a CPLEX solver in the general algebraic modelling system (GAMS) on a redesigned 48-bus DSN. The simulated initial expected load loss of 39% of 35 MW was decreased to 35.34%, and then to 30.71% with the use of hybrid statistical DSN’s FC-MCS-SCENRED, and the BN-DSN predictive models. These results were validated using the Electrical transient analyzer program (ETAP). This study confirmed that the BN-DSN predictive model is a better operational planning tool compared to hybrid statistical DSN’s line FC-MCS-SCENRED predictive model.

The maximum and minimum real power output of the DGs at bus ''x''. S w(t,l) The-binary variable = 1, for a closed switch on line set ''l,'' otherwise = 0 at a time ''t''. Ŕ t ′ ,l For a repairable line ''l'' status at time ''t'', The binary variable = 1. Otherwise, 0. ς sw(t,l) Consider that the switch status changes for line ''l'' at a time ''t'', the binary variable = 1 δ t, x If the DG at bus-bar ''x'', at a time ''t'' is planned then the binary variable = 1, otherwise = 0. V t, x The magnitude of the voltage at bus ''x'', time, ''t''. V t,y The magnitude of the voltage at bus ''y'', time, ''t''. γ x,y,t If bus ''y'' is the parent of bus ''x'', at a time ''t'', binary variable = 1.

P LS t,x
Interrupted real power at bus ''x'', in MW

I. INTRODUCTION
A reliable electricity supply supports human existence, societal growth, and civilization globally. Regrettably, power system network (PSN) infrastructures are usually subjected to damage during high impact low likelihood (HILL) events. The HILL events could either be extreme weather events that are predictable and unpredictable or human-induced power system operational errors. These two major sources of fault usually result in cascaded power outages with a lot of loss of revenue. From the published literature, the study of the causes of the widespread power blackout conducted in the United States of America (U.S.A) in 2008, showed that between 1984 and the year 2006, 933 events resulted in largescale power outages. 44% of these events were caused by natural events such as hurricane events [1], [2]. The United States Department of Energy (US-DOE) data showed that out of 1,333 power system network damages recorded between 1992 and the year 2011, almost 78% were caused by extreme weather events [3]. Hurricane event-induced power outages on the DSN are very common around the globe these days due to global warming [4]. For example, on the 11th of June 2018, hurricane Ewiniar dumped very heavy rainfall in Vietnam along the western coast of the South China sea. In this event, fourteen (14) people died. Loss of social infrastructures worth ¥5.19 billion ($749 million) was recorded [5], [6]. Over the past 50 years, it has been documented that more than 30 tropical cyclones have affected the South African mainland [7]. On the 29th of August 2021, a catastrophic category-4 Hurricane Ida hit Louisiana tearing off the basic infrastructures like hospital roofs, the DSN, etc. In this incident, millions of people were thrown into a total power blackout. Over 1 million electricity end-users experienced a total blackout in New Orleans, Louisiana and Mississippi, with a record of two (2) lives lost in Louisiana [8]. In a report presented on CNN by Rachel Ramirez, it was noted that researchers raised alarm on increased large-scale power outages due to the extreme weather around Texas, Michigan and California. About 1,500 large-scale power outages were recorded from the year 2000 to 2022 [9]. The July 8th 2022 flood event in Nigeria threw several people living around Lekki, Ikoyi, and Victoria Island of Lagos State into a total blackout for several days due to the power system sub-station and power equipment that was submerged by the flood [10]. The Africa report states that ''there is hardly any work done on improving the institution that works on disaster management in Nigeria'' [11]. This was evident as the report of 29th October 2022, indicated that twenty-seven (27) out of thirty-six (36) states in Nigeria have been hit by the flood. In this event, six hundred and three (603) people died. Two million-five hundred and four thousand, and ninety-five (2,504,095) people were affected. Two thousand four hundred and seven (2,407) people were injured, while one-million, three hundred and two thousand, five hundred and eighty-nine (1,302,589) people were displaced [11]. In view of this, this article is motivated to focus on improving the operational resilience of the DSN against predictable extreme weather events like hurricanes. The goal is to lessen the expected load loss, and the impact of the protracted widespread power outages proactively, leveraging the proposed DSN predictive algorithms in this paper for the DSN proactive operational planning decision-making.
A comprehensive literature review of the steps and strategies utilized to achieve power system resilience improvement against HILL events such as hurricanes was presented in [12] and [13]. These papers suggested the need to change the reactive operational resilience improvement techniques proposed by many researchers in assessing the resilience improvement of the DSN against hurricane events, to a more defensive proactive operational resilience improvement approach [14]. In Omogoye et al., [15], [16], it was revealed that the prospective statistical regression methods (such as the generalized linear model (GLM), generalized additive model (GAM), system tree-approach mining models (STMM),) and the power system topology (PST)-based resilience evaluation approach (such as the predictive DSN line's FC-MCS) have been utilized to assess hurricane-caused DSN line faults in the past. The GLM, GAM, and STMM approaches are grouped as system-level damage predictive algorithms [17]. These approaches are appropriate for long-term PSN infrastructural planning. However, an increase in PSN line fault prediction approximation error limits their application for the shortterm operational resilience improvement planning against predicted oncoming hurricane events. Similarly, the power system topology (PST)-based resilience evaluation approach which includes a combination of a statistical DSN line's FC-MCS predictive model is also known to introduce high computational complexity due to the use of 1000 or more predicted system line fault scenarios for a proactive stochastic system optimization problem formulation and simulation as demonstrated in [15], [16], and [18]. It is worth noting that in all the prospective methods aforementioned above, the uncertain nature of the hurricane event on the PSN was not captured. Therefore, to address the problem of neglect of the non-stationary nature of hurricane events on the PSN when modelling the DSN line fault predictive algorithm, a machine learning dynamic BN-DSN line fault predictive algorithm was proposed in [19]. The proposed BN-DSN predictive model considered fully the uncertainties of the hurricane events on the modified IEEE 14-bus DSN lines, to predict the DSN line faults. However, the BN-DSN line prediction validation model was not developed in [19]. The improved hybrid statistical DSN line's FC-MCS-SCENRED predictive algorithm that validates the proposed BN-DSN predictive model in [19] was presented in [20]. The study where both the proposed BN-DSN and hybrid statistical DSN line's FC-MCS-SCENRED predictive algorithms were developed using IEEE 15-bus to determine their prediction accuracy can be found in [21]. The study presented the full analysis of the two predictive algorithms' accuracy and concluded that the proposed BN-DSN predictive model outperformed the combined statistical DSN's line FC-MCS-SCENRED model. However, the proposed model in [21] was never tested with a proactive system optimization model for impact assessment as regards the reduction in the expected load loss. Other literature such as [16], [19], and [22] also supported the proposed BN-DSN predictive model as a better operational planning decision-making tool for the distribution system operators (DSO) compared to the combined statistical DSN's line FC-MCS-SCENRED predictive algorithm. Generally, studies to corroborate or support this claim are largely unexplored and hence, presents a research gap.
To contribute to knowledge from this perspective, this study presents a comparative performance-based analysis between the proposed BN-DSN and combined statistical DSN lines FC-MCS-SCENRED predictive algorithms, to determine the best DSN predictive algorithm that can reduce the expected load loss proactively from unstoppable forecasted oncoming hurricane events. This type of system operation planning tool would enhance the best proactive operational planning decision-making by the DSO to strengthen the DSN operational resilience planning against the forecasted oncoming hurricane event in the future. It will also help the DSO to assess the power system network risk and vulnerability against any future predictable extreme weather events since the extreme weather events will not stop and cannot be controlled.
The novelties introduced in the study, include the development of different predictable weather-dependent DSN line fault predictive algorithms, conducting performance-based tests on them for proactive operational resilience enhancement against hurricane events, and integrating the best predictive algorithm in a proactive system optimization model to decrease the expected load loss. The objectives are to: reduce the revenue deficit for the power utility companies; and reduce the protracted power outages of electricity end-users FIGURE 1. Redesigned 48-bus DSN one-line diagram [18], [24].
to enhance electricity end-users' survivability after a hurricane event in the future. Hence, the operational resilience of the DSN lines against predictable extreme weather events like a hurricane is enhanced. This goal motivates this study.
The remaining sections of this paper are arranged as follows: Section II presents research data and the methodologies. In section III, the comparative analysis of the efficiency of the two proposed system line fault predictive algorithms is presented. Section IV discusses the pre-hurricane event DSN reconfiguration and resource planning for the system's operational resilience improvement. Section V gives detailed information on the system optimization simulation results analysis, while section VI, concludes the paper.

A. DESCRIPTION OF RESEARCH DATA, MODIFIED 48-BUS DSN, AND SOFTWARE TOOLS
Hurricane event is the natural disaster considered in this article. This is due to its harsh impact on the DSN [23]. The past hurricane events' wind speed intensity measured in meters per second (m/s) called ''typhoon_data_mat'' that was collected between 1949 to 2019 from Shenzhen region in China was utilised to train the prospective combined statistical DSN line's FC-MCS-SCENRED and the BN-DSN line fault predictive algorithms in this study. The predicted oncoming hurricane Ewiniar data in [19], was utilized to perform the DSN line fault prediction. The small size of the redesigned 48-bus DSN was selected for this research investigation because of the experienced complexity associated with the BN-DSN design calculation using a larger network. Fig.1 shows a region-bound one-line diagram of the redesigned 48-bus DSN with a voltage capacity of 200 kV, one (1) transmission bus bar, four (4) distribution feeders (DF1, DF2, DF3, and DF4) with an approximate thermal capacity of 10 MVA, three (3) assumed normally open remote-controlled switches located on buses 9, 21, and 31, and six (6) assumed normally closed remote-controlled switches located on buses  [18], [24].   [24]. The system MVA base is 10 MVA and the minimum and maximum levels of voltage magnitude at the DSN bus are 0.95 and 1.05 p. u, respectively [24]. Table 1 presents the redesigned 48-bus DSN's daily load demand profile. The distributed energy resources (DER) unit locations and their ratings, the daily PV and WTs power output data can be found in [24]. The DSN is assumed to have nine (9) remote-controlled switches (RCSs) located on buses 8, 9, 18, 21, 25, 30, 31, 38, and 42. Software tools which include Microsoft-word package, Excel spreadsheet, Mendeley reference manager, R-software, GAMS, and ETAP were employed for this research investigation. A personal computer with an Intel core i3-6100UCPU@2.30 GHz processor with 8 GB RAM was utilized to perform all simulations.

B. RESEARCH METHODOLOGIES
The prospective procedural steps to perform the pre-hurricane DSN optimization to minimize the expected load loss using the DSN line faults prediction scenarios from the two prospective hurricane-induced DSN line fault predictive algorithms are presented in Fig. 2.  Figure 3 shows the overview of the proposed BN-DSN predictive algorithm flow chart. The detailed step-by-step design procedures for the proposed BN-DSN predictive model can be found in [21]. The prediction accuracy analysis is the focus of the paper. The study demonstrated that the proposed  [20], [21]. In [25], the proposed BN-DSN was also applied to a 48-bus DSN to forecast the DSN line's fault scenario leveraging the predicted oncoming hurricane Ewiniar data. The DSN line's damage scenario prediction enhanced the DSO's situational awareness to perform a cost-effective system operational planning against a predictable hurricane event. Therefore, for the readers' clarity, with no intention of repeating an already established concept in our previous papers, the procedural steps for the development of the proposed BN-DSN line fault predictive algorithm were improved and abstractly presented in this section.
To evaluate the reliability or failure probabilities of the DSN components, one ought to consider the reality that the DSN lines can fail due to one or more DSN components' failure. In this paper, it is considered that the power outage in the DSN in a particular area can lead to power supply interruption in another area. This implies that power supply service interruption may be caused by the failure of the DSN components such as poles and overhead wires, due to the dynamic impact of hurricane wind speed intensity across the DSN. Therefore, the study of the DSN pole damage as DSN line fault ''causalities'' are based on the elasticity perseverance of the DSN poles amid hurricane wind speed intensity (HS) and the DSN poles' foundation's quality that provides resistance to the pole stretch against HS. Due to the unavailability of all the required data about the DSN, the conceivable overhead electrical cable structures are utilized to depict the DSN topology structural reliability as shown in Fig. 4 [26]. Therefore, this proposed DSN line fault predictive model gives a sound premise to the BN-DSN line fault prediction analysis.
Considering the flexural failure points of the DSN lines shown in Fig. 4, both the pole structure and the overhead conductors exposed to the hurricane event would experience the stress of the HS that can possibly result to a pole bending moment, to the ground. In this context, the DSN pole failure under a hurricane event can be expressed as the peak bending moment (BM) shown in (1): where M p is the DSN pole bending moment caused by HS, M w represents HS induced bending moment of the DSN's conductors, and W w represents the conductor's span length.
The inside strands of the DSN poles are packed and the external filaments are stretched out because of the pole's base bending moment. In case the tensile stress of the extended external fibre surpasses the most extreme pole crack stretch, at that point the DSN pole will be broken. The breaking point state capacity of the flexural failure mode G(x) is introduced in (2) [26].
where R is the resistance capacity, W is the wind load, δ r is the mean modulus of rupture (MOR) of the DSN poles (r), and ground line (δ g ) is the tensile stress of the DSN pole at the ground line, M g is the bending moment at the ground line, Z represents pole section modulus, and D represents the diameter of the pole at the ground line. The detailed modelling of the designated fibre stress due to HS on various wooden poles in the DSN has been proposed in [27], while the yardstick for quantifying the resilience of the power utility DSN's poles is proposed in [28]. In the BN-DSN model, an all-inclusive CPD related to hurricane event forecast is utilized to define the DSN line fault forecast problem formulation. The failure probabilities (FP) of the DSN lines under hurricane events can be represented by (3) [29], [30].
In the development of the BN-DSN line fault predictive algorithm, two factors were contemplated to be causing the power outages on the DSN lines whenever a hurricane event struck. These factors include the hurricane wind speed (HS) measured in meters per second (m/s), and the situation of the upper feeder pole (UFP) distributing power to the lower feeder pole (LFP) lines. This study utilized these factors to design the proposed BN-directed acyclic graph (BN-DAG) shown in Fig. 5, based on graph theory concept.
A graph theory technique represented by G (N, B) was utilized to evaluate the condition of the 48-bus DSN's lines status when a hurricane event struck. G is the graph, N depicts the DSN's parent (UFP and HS) and child nodes (SL) condition respectively. B on the other hand, represents the DSN link between the parents' nodes and the child node. To estimate the 48-bus DSN line fault's probabilities using the conditional probability distribution (CPD) of Shenzhen region of China, the need to put into consideration the influence of the two independent random and continuous variables of the parents' nodes become imperative. The condition of the two parents' nodes directly affects the child node's status (S L ) as shown in Fig. 5. Therefore, the general term to describe the regional CPD of the hurricane-prone zone under study, that models the DSN line fault prediction is presented in (3) as proposed in [29] and [30], respectively.
where P 0 Fl is the likelihood that the DSN line will be broken or not broken when the hurricane events struck the grid ''l'' is a subset DSN line of the universal DSN line set ''L''. S L depicts the DSN line's condition after the hurricane event (that is S L = 0, or S L = 1). HS represents the hurricane event wind speed data. Therefore, if the probability distribution function (PDF) of the hurricane-prone region and the predicted approaching hurricane data are provided by the weather station, the BN-DSN-DAG model in Fig. 5, can be put in place to predict the maximum likelihood of the DSN line fault. The mathematical expression for this formulation is shown in (4) [29], [30].
where Nodes BN = {n i }, i ∈ [1, N ] is the set of random variables described in Fig. 5. Line BN = n i , n j , i ̸ = j, represents the edge set, that is, the grid's branch of the proposed BN predictive model. Lines BN depicts the main causes of the DSN line fault. The two parents and child node relationship are described by the nodes (n i ), and edges (nj). For instance, consider n i , where all its parent nodes and child node are classified as the set of π UFP,HS,SL (n i ) , therefore a parent node with a known value can be defined as an independent node in a BN-DSN-DAG model. Similarly, the dependent node (n j ) that utilizes the hurricane-prone region CPD, represented by θ BN , is also defined in the BN-DSN-DAG model, given that the values of all its parent nodes are provided. The DSN-CPD mathematical expression for the proposed BN-DSN line fault model is presented in (5) [29], [30].
In this study, it was assumed that if the dynamic HS across the 48-bus DSN is above 65 m/s (category 4-hurricane event) and it is sustained for more than 60 seconds across the DSN, any DSN pole hit with such HS will be broken, otherwise it will not be broken. In the same vein, if UFP line of the DSN is damaged, no power would be supplied to the LPF lines. These two constraints were used to model the DSN line status after a hurricane event. This relationship is presented in (6).
To model the BN-CPD between the HS and the DSN-UFP lines, the past recorded and stored hurricane event data of the Shenzhen region in China was utilized [19]. The goal of the proposed BN-CPD is to generate the DSN line fault probabilities using the CPD of the region under study to model the DSN line damage or not damage conditions. Therefore, using the proposed BN-DSN-DAG model structure in Fig. 5, the 48-bus DSN line fault under the predicted oncoming HS is modeled as presented in Fig. 6. In Fig. 6, the DSN line 1, (SL1), is the root node (i.e., the transmission bus). SL2, SL10, SL22, and SL34 represent the power supplying distribution feeders (DFs) 1, 2, 3, and 4 respectively. The BN-DSN predictive model mimics the existing relationship between the 48-bus DSN's poles and HS, taking into account the conditional dependence between the DSN parents and child nodes via the DSN branches. This is translated as BN-DSN structure using the region's PDF and the CPD approaches shown in Table 3.
In Table 3, P (SL2 = LB or LNB|HS2) represents the probability that the SL2 ''will not be broken'' or ''will not be broken'' given that it is struck by hurricane wind speed intensity 2 (HS2). In the same vein, P (SL3 = LB or LNB|HS3, SL2) means the probability that the SL3 ''will be broken'' or   ''will not be broken'' provided that, it is connected to SL2 when it was hit by the HS3. Using this approach, the PDF is computed for each pole in the 48-bus DSN as presented in Table 3. The derived 48-bus DSN poles' failure probabilities under hurricane event are presented in Table 3, and plotted in Fig. 7 respectively. The generated 48-bus DSN pole failure probabilities in Table 3, are then compared with the random numbers generated ranging between 0 and 1. The decision-making condition for the predictive probabilistic failure model states that, if the 48-bus DSN line failure probability of each line is less than the random number, the DSN pole is not broken (LNB), otherwise, the DSN pole is broken (LB). This approach is established using Monte Carlo Simulation (MCS) algorithm to generate the 48-bus DSN line fault scenarios data. The data is used alongside with the historical hurricane event data to train the proposed BN-DSN predictive model for future DSN line fault prediction. The mathematical expression for the data set D HS is presented in (7).
To ascertain the CPD of the 48-bus DSN lines, the softmax approach presented in (8) is employed to handle both the discrete and continuous parents' nodes of Figs. 5 and 6 [31].
represents the relative weight of the normal vector for the HS within the 48-bus DSN, and b l V depicts softmax offset. The relative weight and the biases/offset of the softmax in (8) is defined by (9) as indeterminate parameters of the 48-bus DSN line ''l'' [31].
For every DSN line ''l,'' in the 48-bus DSN, ∅ l determines their CPD. Also, ∅ l can be expressed in term of the BN-DSN predictive model undefined parameters as provided in (10): The BN-DSN predictive model's training goal is to determine the DSN's optimal CPDs represented by ⊖ * BN . The optimal learning process is carried out using the offline stored data set in (7). The 48-bus DSN's optimal learning problem formulation is presented in (11) [29], [30].
The criterion of (11) can be learned by the use of a gradientdescent algorithm, given that the actual mastering rate of η ∈ (0,1) are provided and the initial value of the optimal CPDs of φ (0) is known. Subsequently, after a few simulations, the parameter of the BN-DSN predictive model will accomplish convergence based on the guideline of maximum probability assessment of the preparing data set. At this stage, the complete proposed BN-DSN line fault predictive algorithm is presented as a joint probability distribution (JPD) using the 48-bus DSN nodes (Nodes BN ). The BN-DAG random variables from Fig. 5 are presented, using (13), [29], [30].
where S = {S L l ∈ [1, L]}, S L represents the vector quantity with binary values of 0 and 1 showing the whole system line status after the hurricane event. S represents the whole scenarios of the 48-bus DSN line status. Therefore, the CPD of the random variables for the 48-bus DSN line status (S) is given by (14), [29], [30].
where HS = [HS 1 . . . , HS l . . . , HS L ] is the set of hurricane speeds across the 48-bus DSN. Equation (14) is assessed with the hurricane Ewiniar prediction data to determine the likelihood of the hurricane event occurrence and the range of the 48-bus DSN line damage. Therefore, the BN-DSN line fault model estimates the DSN line status based on the rule of highest probability of the DSN line fault when the forecasted oncoming hurricane event struck the grid as given in (15), [29], [30].
where S * represents the 48-DSN line fault range. S * is also the DSN line fault prediction results presented in terms of LB or LNB in this study or better presented as S l = 0 or S l = 1, that is 0 means line damage, and 1 means otherwise. The proposed BN-DSN predictive model is implemented on  the modified 48-bus DSN using the CPD of the proposed BN-DSN-DAG predictive algorithm. The hurricane-induced power system line fault prediction was carried out using Hurricane Ewiniar data [32], with its corresponding simulated system line fault scenario. The BN-DSN predictive model program written with MATLAB toolbox by Murphy in [31] and [33] was studied and modified using R program [34], to suit our BN-DSN line faults prediction goal. The detailed algorithm flow chart for the BN-DSN line fault simulation is presented in Fig. 8. The resulting 48-bus DSN line fault prediction under hurricane Ewiniar is presented in Table 3.  (1) and (2) respectively. Having stated this fact, our past work that presents the detailed procedural steps for the development of the proposed combined statistical DSN line's FC-MCS-SCENRED predictive model can be found in [20]. However, for readers' convenience, this concept is abstractly presented in this section using a modified 48-bus DSN. The DSN's wooden pole and conductor are considered for this investigation. The first task is to establish the risk of a wooden pole's failure under HS and scale this risk over the DSN configuration as displayed in Fig. 4. The DSN maximum bending moment under HS is defined by (1) [26]. According to the rule of structural reliability in Civil Engineering field, which states that if electric pole cracks caused by an extreme weather condition are distributed within the component, as component's failure probabilities, the crack propagation of the DSN pole can eventually cause the electric line to fail under extreme weather conditions causing a large-scale power blackout [35]. Therefore, the DSN pole failure probabilities under hurricane events were modelled as presented in (16) [36], [37].
where, P F (σ ) is the electric pole probability of failure, as a function of failure stress(σ ) of HS. σ 0 , represents the reference pole stretch under HS, and m represents the statistical Weibull modulus. Both (σ 0 , m) are calculated from the available historical hurricane data. Given historical hurricane Ewiniar data, the zonal HS, PDF, CDF, and the redesigned 48-bus DSN poles failure probabilities are calculated by rewritten (16) to obtain (17) [38].
By taking the log of both sides of (17), (18) is obtained.
Equation (18) is presented as a straight-line equation, in the form of y = mx + b shown in (19). The breakdown is presented in Table 4.
It was modeled such that the HS that passed through the 48bus DSN began to stress the DSN's poles from hurricane wind speed of 24.85 ms −1 to 79.59 ms −1 , and at 80 ms −1 and above, the pole starts to fail. The corresponding DSN line's failure probabilities is presented in Fig. 9, having duly considered the impact of hurricane wind speed's PDF and CPD of the region under study. The Statistical Weibull parameters of slope m = 3.862 and σ O = 60.958, derived from (16) were used for graph plotting shown in Fig. 9. Using (16), the redesigned 48-bus DSN's poles failure probabilities due to HS are obtained. The data is then compared with their corresponding generated random numbers (r n ) ranging between 0 and 1 (r n∼ 0, 1), (See Table 5). The rule that determines the condition of the redesigned 48-bus DSN line status as presented in (20), states that, if each generated random number for each DSN line is compared with its corresponding DSN's line failure probabilities under a hurricane event, and the random number is less than the DSN pole's failure probabilities, the line is damaged ''1'', otherwise the line is not damaged ''0'' [39]. But in this article, ''0'', is used to represent line fault while ''1'' is used to represent DSN line, not damage. This is the condition used to model this combined statistical DSN line's FC-MCS-SCENRED predictive algorithm.
where P L F H HS l = 0 is the likelihood that the redesigned 48-bus DSN lines will fail or be broken, whereas when P L F (H HS l ) = 1 is the likelihood that the redesigned 48-bus DSN lines will not fail or will not be broken.
In the same vein, due to the need to take into account the probabilistic nature of the hurricane event across the redesigned 48-bus DSN, the MCS model is employed to forecast the possible DSN line fault scenarios due to hurricane Ewiniar [40]. The MCS has been employed in many studies in the past without defining proper stopping criteria. This usually affects its prediction accuracy since there are no universally accepted stopping criteria currently [22]. However, some studies such as [18] and [39], in power system resilience, claimed that 1000 DSN line fault scenarios prediction from the MCS model is enough to guarantee power system line fault prediction accuracy under extreme weather events like hurricane events. Therefore, 1000 MCS line fault prediction is adopted in this study. In this view, the DSN line fault prediction was simulated using the redesigned 48-bus DSN, on an excel spreadsheet. The samples of the redesigned 48-bus DSN line fault prediction from the DSN line's FC-MCS model is shown in Table 5. Each DSN line fault prediction scenario represents the system line status and the DSN's response to the hurricane events. It should be noted that the combined statistic stochastic system optimization simulation result's accuracy largely depends on the higher number of the DSN line fault scenarios generated from the predictive statistical DSN line's FC-MCS model. Alas, these higher DSN line fault scenarios generated usually result in computational high time costs when the generated large numbers of system line fault scenarios are used to perform stochastic system optimization simulation. In this context, one of the ways around this problem of computational time cost is to apply the scenario reduction (SCENRED) algorithm to the modified 48-bus DSN's 1000 line fault prediction scenarios. This is to reduce the generated system line fault scenarios to a smaller desired number of the DSN line fault scenarios without jeopardizing the precision of the predicted DSN line damages scenarios [20]. The reduced DSN line fault scenarios are then used to perform the pre-hurricane events stochastic system optimization [41]. The developed SCENRED program in GAMS [42], was applied to the modified 48-bus DSN line damages scenarios predicted from combined statistical DSN's line FC-MCS model, to reduce its predicted 1000-line fault scenarios to 10 with their corresponding optimum probabilities of system line fault for each reduced system line fault scenario generated. See Table 6 for the 10 reduced DSN line fault scenarios.

III. DISCUSSION ON THE PROPOSED BN-DSN AND HYBRID STATISTICAL DSN'S LINE FAULT PREDICTION MODELS
The efficiency of the proposed predictive BN-DSN and the combined statistical DSN line's FC-MCS-SCENRED predictive models under hurricane events was studied through the DSN line fault simulation on the 48-bus DSN. In both cases, the algorithm solvers depend on adaptive search techniques, particularly, the Branch-and-Bound method [43]. So, the closeness in the prediction time and memory as presented in Table 7, is not surprising because of the hybridization of SCENRED algorithm to the combined statistical DSN line's FC-MCS model. The improved statistical predictive algorithm optimally reduces the system line fault prediction to a small sizable number without jeopardizing the system line fault prediction accuracy [41], [44]. Comparing the DSN line fault prediction time and computer memory usage in both predictive algorithms as shown in Table 7   FC-MCS-SCENRED predictive model. The accuracy of both the proposed DSN line fault predictive algorithms can be confirmed as the scenario 7 of Table 6, which recorded the highest probability of DSN line fault scenario to occur, if a hurricane event struck the DSN, matched the single line fault prediction scenario in Table 3. The detailed explanation of prediction accuracy analysis can be found in [21] and [25].
The predicted DSN line's fault scenarios from both the predictive algorithms under hurricane event are leveraged, by the DSO as power system resilience proactive operational planning tool to decrease the expected load loss. In this context, the pre-hurricane event DSN optimization problems were formulated based on the two predicted DSN line fault scenarios. The impact of the two predictive algorithms was quantified and compared using minimization of the expected load loss as the resilience metric. The modelling of the pre-event DSN optimization is detailed in the next section.

IV. PRE-HURRICANE EVENT DSN RECONFIGURATION AND RESOURCES PLANNING FOR RESILIENCE IMPROVEMENT
The review of system network reconfiguration methodologies presented in [17], [22], and [45] provide a guide in choosing the most appropriate DSN reconfiguration optimization problem formulation approach employed in this study. Morelato and Monticelli [46], revealed that due to the magnitude and non-linear nature of the DSN, a combination of system optimization and heuristic approaches could not achieve an optimal solution [46]. It then suggested that, since it is possible to linearize the pair of objective function and the system  operational constraints, to change the non-linear nature of the DSN reconfiguration problem to a MILP system optimization [45], [47], the MILP problem formulation approach was a choice employed in this paper. The prospective optimal DSN reconfiguration and resource allocation MILP problem formulations were employed to conduct a performance-based comparative study to determine the prospective predictive algorithm that is most appropriate to execute pre-hurricane event DSN operational planning in the future. The formulated MILP optimization problems in the two cases, were solved using a CPLEX optimizer in GAMS. The DSN optimization solution proposed in this paper drew its inspiration from the pre-hurricane event DSN optimization problem formulated as the crew dispatch and transportation problem in [18], the pre-and post-event system optimization under hurricane event in [48] and [49], and the prospective two-stages system optimization under hurricane event in [50] respectively.  Table 3 in (22) and all the reduced 10 predicted system line fault scenarios in Table 6 in (21). The prospective formulated system optimization problems are formulated as a MILP problem in (21) and (22)  Equations (21) and (22) are subjected to the following operational constraints one after the other. To ensure balanced power flow, the real and reactive power flow within the redesigned 48-bus DSN at a time ''t'' are modelled as (23) and (24), where P D t,x and Q D t,x are the active and reactive power demand at the bus ''x'' at a time ''t''. P DG out@t,x and Q DG out@t,x , P PV out@t,x and Q PV out@t,x , and P WT out@t,x and Q WT out@t,x are the active and reactive power output of the distributed energy resources injected at the bus ''x'', time ''t''. P Sub t,x andQ Sub t,x are variables that indicate the active and reactive power injected from the VOLUME 11, 2023 transmission line, while Constraints (26) and (27) are based on the Distflow model of a linear AC power flow proposed in [51]. Constraints (26) and (27) are relaxed through the big-M technique [52], provided all other attached lines to the bus ''x'' are in the open state to adhere to network operational constraints [53]. An explanation of how the big-M technique is utilized to relax the constraints in (26) and (27) can be found in [52].

Vt,y−Vt,x+
Rl P Vt,y−Vt,x+ Rl P In making sure that the magnitude of voltage experienced at all buses of the redesigned 48-bus DSN is within the normal operating range, constraint (28) is employed.
To ensure that the maximum capacity limit of complex power flowing in a line's is within the line operating limit, (29) is used. Also, to maintain the linearity of the model, the special-ordered-sets-of-type 2 (SOS2) approach is employed to linearize (29), [54].
The power output ranges for the DGs, the PV, and the WTs are modelled with the consideration that the PV and WTs are environmental time-varying solar irradiation and wind speed dependent energy-generating components. Therefore, this study assumed that the WTs and PV have a constant power factor, and that the power output sources are P-Q sources. Constraints (30) and (31) Furthermore, (32) and (33) ensure that the output power of the photovoltaic panel and the wind turbine units did not exceed their respective maximum power output limits.
To ensure that the DSN is kept in a radial configuration, the constraint proposed in [55], the spanning-tree approach for the power system network analysis is employed for (34), (35), and (36) respectively. The constraints make sure that line ''l ′′ at a time ''t'' in the spanning tree structure is kept in a radial configuration. That is, S w(t,l) = 1, if either of γ x,y,t orγ y,x,t = 1. This implies that when γ y,x,t = 1, ''y'' represents the parent bus of ''x'' at a time ''t'', and when γ x,y,t = 1 ''x'' is the parent bus of ''y'' at a time ''t'' respectively.
To put in check the number of times the RCS status changes, constraint (37) is introduced. If the switch status changes for line ''l'' at a time ''t'', the binary variable RCS sw(t,l) will be 1, otherwise 0.
The change in switch status can be detected using, constraints (38) and (39) The time limit to operate the RCSs before hurricane event can be mathematically formulated as (40) and (41) respectively. That is the RCS actions to optimally reconfigure the network modelled in (40) must be performed under the bound of (41), otherwise the pre-hurricane event optimal DSN and resource scheduling will be defeated.
The DSN optimization problems formulated in MILP method in this section, are solved using a CPLEX solver in the GAMS, making use of a branch and bound algorithm. The solution encourages the deployment of specially ordered sets of variables SOS1, SOS2, semi-continuous, and semi-integer variables proposed in [56]. The algorithm also provides a detailed technique to solve a linearized MILP problem by modifying the constraints in the state-space search approach [43]. It is worth to note that the following assumptions were made prior to the formulation of the pre-hurricane DSN optimization problems: 1) All switches are remote-controlled switches.
2) The distributed energy resources (DERs) have been optimally placed in the redesigned 48-bus DSN [24]. Therefore, problems related to voltage fluctuation from the DERs are ignored in this paper. 3) It was assumed that the forecasted oncoming hurricane event is to hit the 48-bus DSN in 60 minutes ahead.

B. THE MILP SIMULATION PLATFORM
The MILP model is fed with the detailed redesigned 48-bus DSN data, remotely controlled switches locations, and predicted DSN line-fault scenarios from both the prospective predictive algorithms. The proposed formulated MILP optimization simulation constraints described in Section IV-A, are used to perform the two cases of pre-hurricane event DSN optimization simulations. The next section describes the system resilience metrics adopted for the simulation results analysis.

C. THE SYSTEM RESILIENCE METRICS
The resilience metrics proposed in [39], shown in Table 8 were adopted to analyze the redesigned 48-bus DSN resilience simulation results since there is no universally accepted standard power system resilience metric at the time of compilation of this study report [22], [57].

V. PRE-HURRICANE DSN SIMULATION RESULTS ANALYSIS A. EVALUATION OF THE EFFECTIVENESS OF THE PROPOSED BN-DSN PREDICTIVE MODEL AND HYBRID DSN'S LINE FC-MCS-SCENRED MODEL
To investigate the impact of the proposed system network line fault predictive algorithms as regards the DSN's operational resilience enhancement against hurricane events, three simulation case studies were considered. The simulation results are discussed in the context of the DSN's operational resilience improvement against predicted oncoming hurricane events. The simulated case scenarios include: 1) A simulation scenario where no proactive DSN reconfiguration and resource planning operation is performed against the forecasted oncoming hurricane event. In this case, the most probable DSN line's FC-MCS-SCENRED line fault scenario of Table 6 was taken to be the actual system network line fault scenario. This is because the prospective BN-DSN predictive line fault scenario in Table 3 matched this DSN line fault scenario 7, of Table 6, that recorded the maximum probability of DSN line fault due to the approaching hurricane event. Therefore, the system line fault scenario is used to simulate the active power flow on the grid to determine the expected active load loss that would be experienced. The simulated expected active load loss in this case is then used as a reference active load loss for further active load loss reduction analysis when the proactive DSN's reconfiguration and resource planning are applied using the simulated damage scenarios from the two prospective DSN predictive algorithms. 2) A simulation scenario where the pre-hurricane event system optimization is performed based on the combined statistical DSN line's FC-MCS-SCENRED prediction results of Table 6. 3) A simulation scenario where the pre-hurricane event system optimization is performed based on the BN-DSN predicted line fault scenario of Table 3.
Since the forecasted oncoming hurricane event was due to hit the 48-bus DSN in 60 minutes ahead, the DSO performs the pre-hurricane event prediction simulation to determine the DSN's most probable line fault scenarios using the two proposed predictive algorithms in this study. The analysis of the simulation results is reported in the next section.

B. ANALYSIS OF SIMULATION RESULTS OF THE PRE-HURRICANE EVENT DSN EXPECTED LOAD LOSS BASED ON THE PREDICTED MOST PROBABLE DSN LINE FAULT SCENARIO
Initially, before the hurricane event, the redesigned 48-bus DSN's peak active load demand of 3.5 p. u (35 MW) was met between the 19th and 22nd hours of the day. See Table 1. Considering Table 6, scenario 7 that matched the damage scenario in Table 3, the DSN fault scenario is regarded as the most probable DSN line fault to be experienced when the hurricane struck. In this case, the expected load shedding was simulated to be 1.365 p. u. This means 13.65 MW of loads would be unserved when the predicted oncoming hurricane event hit the DSN. Therefore, 1.365 p. u (13.65 MW) was used as the benchmark to quantify the operational resilience enhancement that would be achieved, after the application of the pre-hurricane event system optimization based on predicted DSN line fault scenarios in Tables 3 and 6 respectively. The objective function is to minimize the expected load loss via pre-hurricane events' optimal DSN reconfiguration and resource planning, via optimal switching sequence of the RCS operations. The pre-hurricane event system optimization simulation solution was proven optimal with a CPLEX execution time of 1.78 seconds. The MILP optimization solution of 2.263 p. u (22.63 MW) with its absolute and relative optimality gap of zero (0), was obtained. Optimal load shedding of 1.237 p. u (12.37 MW) was recorded over all the possible reduced predicted 10 system line fault scenarios in Table 6. The MILP program generates one optimal DSN reconfiguration switching sequence and resource planning shown in Table 9 and Fig. 10, respectively. These actions are capable of minimizing the expected load shedding  over all the reduced predicted 10 DSN line fault scenarios in Table 6. Table 9 shows the pre-hurricane optimal switching sequence to mitigate the impact of the forecasted oncoming hurricane event on the DSN. Initially, the RCS 1, 3, 5, 7, 8, and 9 are normally closed switches whereas the RCS 2, 4, and 6 are normally open switches. The simulated optimal switching sequence for the first 60 minutes shows that RCS 2 was closed from the initial open state, RCS 3 was opened from the initial closed state, RCS 4 was closed from the initial open state, and RCS 9 was opened from the initial close state position. The total pre-event switching time of 20 minutes was recorded. After 1 hour, switch RCS 3 and 9 were now closed. The simulated optimal pre-hurricane event DSN's DGs planning is presented in Table 10. The corresponding hourly (0-1 hour) DG's active and reactive power planning are presented in Table 11. In Table 12, the active and reactive power output of the scheduled PV and WTs against the oncoming hurricane event is presented. Fig. 11 shows that the voltage magnitudes of all the buses in the redesigned 48-bus DSN are within the DSN normal operating condition voltage magnitudes of 0.95 ≤ |V | ≤ 1.05 respectively when power flow analysis was performed indicating the steady state performance of the system network operating condition.  step. The graphical representation of the switching sequence leading to proactive optimal network reconfiguration and resource scheduling based on the BN-DSN predictive line faults of Table 3 is presented in Fig. 12. Information concerning hourly scheduling of the DSN's DG can be found in Table 14. The hourly DER active power planning is  Table 3.  presented in Table 15. All the BN-DSN based pre-event voltage magnitudes of all buses are within the allowable voltage range of 0.95 ≤ |V| ≤ 1.05 respectively. The analysis of the system optimization simulation results in this paper for the three cases study showed decrease in the initial expected load loss from 39% to 35.34%, and finally to 30.71% respectively (See Table 16). The pre-hurricane event simulation time of 1.78 seconds and 1.56 seconds were recorded for each system optimization simulation using the two predictive models. The simulation time in both cases was less than 2 minutes. confirming the efficacy of the formulated system optimization problems in Section IV-A. The reduction in the expected load loss simulated, simply means a reduction in the expected loss of revenue to the power utility company.   Table 3 was used to model a fault scenario on Electrical Transient Analyzer Program (ETAP) platform. However, ETAP was unable to handle a stochastic system optimization which is possible with Table 6. Fig. 13 shows the modelled redesigned 48-bus DSN network in ETAP. The load restoration data of 2.441 p. u (24.41 MW) reported in Table 17 is similar to the restoration values of 2.425 p. u (24.25 MW) obtained for the pre-hurricane event system optimization simulation result based on the prospective predictive BN-DSN model in Section V(D). Thereby validating the accuracy of the proposed system restoration scheme employed for the system optimal pre-hurricane event network reconfiguration and resource planning in this study.

VI. CONCLUSION
In this article, the pre-hurricane event DSN optimization simulation results have been analyzed to showcase the impact of the two proposed resilience evaluation methods as regards to proactively enhancing the operational resilience planning of the DSN against the predicted oncoming hurricane event.
Firstly, the efficiency of the proposed BN-DSN predictive model was compared with that of the hybrid statistical DSN line's FC-MCS-SCENRED predictive model. It was demonstrated that the stochastic pre-hurricane event system optimization simulation performed using 10 reduced DSN line fault scenarios attained optimality at a CPLEX optimizer time of 1.78 seconds whereas the pre-hurricane event system optimization simulation that was based on the prospective BN-DSN line fault scenario converged at a CPLEX optimizer time of 1.56 seconds. From these simulation results, it can be deduced that if the predictive hybrid statistical DSN line's FC-MCS-SCENRED model's 1000-line fault scenarios were not optimally reduced to 10, using the DSN line FC-MCS-SCENRED model, the 1000 DSN line fault scenarios would have resulted in higher computational time during stochastic system optimization, compared to the system optimization simulation time of 1.56 seconds that was seen in the case of system optimization simulation based on the prospective BN-DSN line fault scenario. Inferentially, the prospective proposed BN-DSN predictive model can significantly enhance the pre-hurricane event DSN operational planning at a reduced system optimization simulation time compared to the combined statistical DSN line's FC-MCS-SCENRED predictive algorithm. The reduction in simulation time will lead to improved electricity end-users' survivability before and after hurricane events, and also reduce the revenue deficit to the power utility companies in future. However, it is very important to note that to implement the state-of-the-art algorithms' solvers for the prospective BN-DSN structure learning problem, an adaptive search technique, precisely the Branch and Bound method, and integer linear programming technique are required. Currently, we noticed that there is no single solver that dominates the other in speed. Therefore, the number of DSN line fault scenarios to be optimized would always determine the system computational time in any case scenario as shown in this study.
Secondly, to assess the DSN operational resilience enhancement level against hurricane events, the predicted most probable DSN line fault scenario was used to perform load flow analysis. 1.365 p. u (13.65 MW) load loss was recorded without any pre-event system optimization scheme. With system optimization based on the prospective hybrid statistical DSN line's FC-MCS-SCENRED predicted DSN line damage scenario 1.365 p. u (13.65 MW) active power supply loss was reduced to 1.237 p. u (12.37 MW) at 1.78 seconds simulation time, with 25 minutes switching time respectively. Similarly, the pre-event system optimization that was carried out based on the prospective BN-DSN predictive model, also reduced the expected load loss from 1.365 p. u (13.65 MW) to 1.075 p. u (10.75 MW) at 1.56 seconds with 20 minutes switching time. The ETAP simulator system optimization results confirmed the correctness of the system optimization results from the BN-DSN system optimization. Therefore, with the reduction in active load shedding, and switching time, it is reasonable to conclude that the predictive BN-DSN line fault model helps the DSO to operationally plan and prepare the redesigned 48-bus DSN against the forecasted oncoming hurricane event, to lessen the hurricane event's impact on the DSN significantly.
Conclusively, the comparative analysis study on the DSN operational resilience enhancement model against HILL events in this article showcases the usefulness of the predictive algorithms as system resilience evaluation techniques. It is demonstrated that the pre-hurricane events system optimization built on the predictive BN-DSN line fault model can greatly enhance the operational resilience of the DSN against hurricane events in the future to reduce the expected active power supply loss when compared with the load loss reduction obtained using hybrid statistical DSN line's FC-MCS-SCENRED predictive model. However, the goal of the DSO is to minimize the load shedding and restore the DSN to normal operational status in the shortest possible time in any case of contingency. In this study, total restoration of the DSN loads is not possible under this research scope. This article did not cover the system components' hardening that bothered on infrastructural resilience improvement planning. In such a situation, one of the reported problems of the infrastructural resilience improvement is insufficient funding or the high cost of system hardening. Therefore, to achieve total power supply restoration in future research, a cost-effective hardening and operational system resilience improvement algorithm is recommended to be developed as a combinatorial optimization approach to solve the probabilistic load restoration problem under both the operational and infrastructural planning approaches. In the same vein, the use of renewable energy resources to serve the loads after a hurricane event is promising in enhancing the system performance. However, the uncertainty associated with DER energy productions because they are weather dependent is a major problem that needs holistic research investigation for proper proactive planning to meet the load demand after hurricane events. It is therefore recommended that this model be developed to improve this current research results in the future. He is currently a Professor with the Department of Electrical Engineering, University of Cape Town, Cape Town, South Africa. His research interests include power system stability, control and optimization, high-voltage direct current modeling, the network integration of renewable energy, the application of artificial intelligence computation to power systems, smart grids, and power system resilience.
Prof. Folly is a member of the Institute of Electrical Engineers of Japan (IEEJ) and a fellow of the South African Institute of Electrical Engineers (SAIEE).
KEHINDE O. AWODELE (Member, IEEE) received the B.Sc. degree in Electronic and Electrical Engineering from the University of Ife (now Obafemi Awolowo University), Nigeria, in 1979, and the M.Sc. degree in Electrical Power and Machines and the PGDM degree from Ahmadu Bello University (ABU), Nigeria, in 1991 and 1999, respectively. She was involved in the electricity meter and circuit breaker manufacturing industry for several years, where she served in various capacities and received series of training with the technical partners' plants in Switzerland, Germany, and Greece, before moving to academics. She was a Lecturer with the Department of Electrical Engineering, Namibia University of Science and Technology, for some years, and later with the Department of Electrical Engineering, University of Cape Town, South Africa. Her research interests include power system reliability and resilience, distributed generation and renewable energy integration, smart grids, protection, demand side management, and power quality.
Mrs. Awodele is a registered Engineer with the Council for the Regulation of Engineering in Nigeria (COREN) and a member of the Nigeria Society of Engineers (NSE).