Distribution systems resilience enhancement via pre-and post-event actions

: Recently, resilience studies have become an indispensable tool for sustainable operation of energy infrastructure. In line with the need, this study presents a mathematical model to enhance resilience level of power distribution systems against natural disasters. The model is designed as a three-stage algorithm according to system operators’ actions. The first stage schedules pre-event actions. At this stage, forecasts about the approaching disaster as well as fragility curves of system components are used to identify failure probability of system components. The failure probabilities are used to trip out the lines as much as possible to defensively operate the distribution network, and advantages of alternatives such as distributed energy resources and normally-open switches are taken to serve critical loads. The second stage is to monitor system operating conditions during the event and identify the status of system components. The third stage mainly focuses on scheduling post-event actions. At this stage, based on real data about different elements of the network, available alternatives are taken to restore as much critical load as possible. To evaluate performance of the model, it is applied to a distribution test system and the results are discussed in detail.


Introduction
In recent years, occurrence of catastrophic events in different countries has caused widespread outages in power systems.Among different events, natural disasters, especially extreme weathers like hurricanes and windstorms, imposed substantial damages on electricity infrastructure [1].It has been estimated that the blackouts caused by adverse weather conditions in the USA have ∼ $25 billion direct economic losses per year [2].Therefore, revisiting power systems' operational and planning studies to enhance the ability of power systems in dealing with such disasters is a necessary task for the systems planners and decision makers.To reach this goal, the concept of power system resilience has been recently introduced to more systematically and quantitatively study ability of the systems to deal with the disasters.According to United Kingdom Cabinet Office, in context of power systems, resilience is the ability to anticipate approaching high-impact lowprobability events, give an appropriate response, and quickly restore from the shock [3].According to customers failure statistics, distribution systems have a significant effect on the interruption of power supply to the customers [4].This highlights the necessity of enhancing resilience of distribution systems in confronting with natural events like hurricanes.
In the literature, many different works have focused on the possible ways and solutions to enhance the resilience of distribution systems.These solutions can be categorised into two different groups, namely, smart operational and hardening strategies [5].In [6], a new approach has been proposed with the intention of improving the resilience of distribution systems through optimal hardening strategy, in order to cope with adverse weather events.The main solutions considered in the paper include vegetation management and upgrading poles based on fragility models and lines failure probability.Amirioun et al. [7] have proposed proactive scheduling for microgrids in advance of flood arrival through preventive measures.To reach this aim, vulnerable components are recognised first.After recognising the vulnerable components and tripping them out, the microgrid is in a safer state.Gao et al. [1] have demonstrated that widespread installation of distributed generators (DGs) can effectively improve the resilience of distribution systems.To achieve this goal, pre-hurricane resource allocation problem has been formulated as a mixedinteger stochastic non-linear program whose simulation results indicate effectiveness of the approach for restoring more critical load in post-event stage.Lei et al. [8] have discussed that prepositioning of mobile emergency generators can decrease interruption durations.
A new procedure for restoring critical loads after the occurrence of natural hazards through dividing system into multiple microgrids via remote controlled switches (RCSs) has been presented in [9].In [10], a new methodology has been developed for networks with multiple connected microgrids to optimise support from adjacent microgrids during emergency conditions.In the same context, a new model has been proposed in [11] to form multiple microgrids following a disaster.A hierarchical outage management scheme in a multiple microgrid environment has been described in [12] wherein, in the first stage, model predictive control algorithm is employed to schedule available resources in each microgrid and afterwards in the second stage, potential power transfer between the microgrids is coordinated.
In [13], a two-stage adaptive robust approach for scheduling microgrids to cope with natural disturbances has been proposed.This approach is involved in day-ahead scheduling of microgrids, which by considering various uncertainties (e.g.real-time market price, islanding event, and net inelastic load uncertainties) tries to attain the best day-ahead scheduling for microgrids in predisturbance stage.The study reported in [14] focuses on resilience enhancement of distribution systems considering hybrid AC/DC microgrids in presence of electric vehicles and renewable distributed generations.Gholami et al. [15] have stated that after hitting of an event, microgrids in islanded mode would be able to supply their local loads in their defined electrical boundaries with taking advantage of available distributed resources.
As a complementary to the past works, this paper proposes a mathematical model to enhance distribution system resilience via some modifying pre-and post-event actions.The model schedules distribution system operators' actions in three different stages.At the first stage, the model optimises pre-event actions according to the forecasts about the approaching hazard and fragility curves of system components [16].At this stage, lines with high failure probability and flowing power are intentionally tripped out as much as possible, and advantages of distributed energy resources and potential manoeuvres in network configuration are taken to serve the total load.In the second stage, the system operator just monitors the situations.At the third stage, the model optimises post-event actions according to the collected real-time data about the system operating conditions.The model distributes crew teams to conduct local actions such as changing the status of manual switches (MSs).In order to achieve practical results, the limited number of crew teams is considered in the study.In addition, travelling duration of the crew teams is considered in conducting both pre-and post-event actions.According to the explanations, key contributions of this work are as follows: • The proposed model optimises both pre-and post-event actions.
This for sure results in lower damage costs compared to the existing models that focus on either pre-or post-event actions.• The model considers a limited number of crew teams and their travelling durations to conduct field actions.• The potential impacts of the event on the transportation system and traffic are captured via different travelling durations before and after the event.• In the case study, effectiveness of pre-and post-event actions is compared via conducting a wide range of sensitivity analyses on key affecting parameters like pre-event time limits, and so on.
The remainder of this paper is organised as follows.The proposed methodology and problem formulation are described in Section 2.
The case study and associated simulation results are given in Section 3. Finally, the relevant conclusions are drawn in Section 4.

Methodology and problem formulation
This section describes implementation procedure of the proposed method.The method consists of three main stages namely preevent scheduling, mid-event condition monitoring, and post-event scheduling.Pre-event actions usually can alleviate consequences of an event following some preventive actions.The actions are reasonably based on forecasts about the event severity as well as system components fragility characteristics.The post-event actions are decided based on the information received by monitoring and data acquisition systems installed all over the distribution systems.Flowchart of the actions taken in different stages of the proposed method is illustrated in Fig. 1.The three stages are thoroughly described in the following subsections.

Pre-event scheduling
In the first stage of the proposed model, to determine the failure probability of system lines, failure probability of conductor and poles should be estimated first.To do so, fragility function concept is used in this paper.In this regard, at first, a proper estimation of the approaching disturbance severity as well as fragility function of the components is needed.The estimations of disturbance severity are usually achieved via weather forecast data, available historical data, and real-time measurements recorded by adjacent stations [17].According to [16], hurricanes and tornados can, respectively, be predicted up to 2 and 72 h in advance.Once an approaching disturbance is predicted, main attributes of the disturbance such as its duration, intensity, and approaching angle are combined with electric components fragility functions to estimate their failure probability.Needless to mention, network lines are the most important and vulnerable components in electric power distribution systems.fragility of these components should be calculated first.The fragility functions can be achieved: (i) experimentally through some practical tests (intentionally falling poles), (ii) analytically using a structural simulation model, (iii) provisionally from statistical analysis of a wide collection of monitored collapses, (iv) using expert judgment, or (v) via a combination of these approaches.Typical fragility curves of distribution lines and poles, taken from [17], are shown in Fig. 2. Interested readers are referred to [2,6,7] and [16][17][18][19][20][21][22][23][24] for different models for estimating line destruction caused by a hurricane.Once failure probability of the components is obtained via mapping the estimated disturbance severity (i.e.tornado speed here) with the associated fragility function, the failure probability of system lines can be achieved as follows: where ρ l w represents failure probability of line l under tornado speed w. ρ l, c w and ρ l, p w are failure probability of the conductor and poles under tornado speed w, respectively.ρ p−ind w designates the failure probability of an individual pole as a function of tornado speed.Finally, N p, l is the total number of poles in the line.In the above, (1) indicates that a system line can go out of service due to failure of the conductor wire or any poles between the associated buses.In this regard, ( 2) is contemplated to calculate combined failure probability of line poles.After determining the failure probability of system lines, network reconfiguration problem is solved in order to avoid wide area interruptions during the event.In this study, it is assumed that system operators always aim at imposing no curtailment to the customers.However, this aim has different aspects in the first and third stages of the model.In the first stage (i.e.pre-event stage), according to the available information about system components and severity of the event, some of the system lines are more likely to trip out in face of the event.So, in this stage, the system operator inclines to serve the system load via redundant paths with lower failure probability, if available, instead of using the lines with higher failure probability.
It is worthwhile to note that no curtailment is allowed at this stage.Thus, this aim can be translated to maximise tripped out lines according to their power flow and failure probability with satisfying the total load.To achieve this aim, system operator takes advantages of tie-lines, manoeuvre points, and distributed energy resources.In the problem, system operational constraints are adhered [25].Since changing status of MSs needs dispatching crew teams, the limited time for pre-event scheduling is vital to be considered in the model.Also, since the time needed for the manual actions may vary by traffic volume on streets, it is necessary to consider the traffic issues in the scheduling.Moreover, it is considered that system operator will dispatch the crew teams to staging locations right after receiving initial signs about a potential disaster in order to be close to switch locations to have a quick response to the disturbance.
In the model, objective function of the pre-event stage (3) is to obtain the best pre-disturbance configuration of the distribution network in advance of a disaster threat via maximising tripped out lines with taking into consideration of their failure probability as well as their power flow The above objective is subject to some technical constraints and limits which are described hereinafter.The first constraint is to ensure active and reactive power balance in all network buses as follows: In the above constraints, the energy provided by the storage units, distributed energy resources, wind turbines, and photovoltaic panels is equal to load minus powers imported from the lines connected to the bus and load curtailment.
The power flowing through a line should be lower than the associated capacity.This condition is presented by (6).The power flows of open lines are modelled using some auxiliary variables as defined in (7).It is worthwhile to mention that the auxiliary variable (i.e.d t, l ) is considered to ensure that voltages of the two buses holding a line are independent if the line is in an open state.The auxiliary variable is zero and has no effect if the line is in a closed state.The active and reactive powers flowing through a line are also calculated via (8) and (9).The auxiliary variables in (8) and ( 9) are to ensure that the power flow of damaged lines is zero The voltage magnitude at all buses must be within the allowable range.This condition is ensured by (10).Additionally, (11) points out the maximum line flow limit from the main grid Constraints ( 12) and ( 13) are considered to model the active and reactive power limitations of distributed generation units.
Moreover, the maximum discharge rate of the storage units is modelled in (14).Furthermore, (15) indicates that output powers of wind turbines could not exceed their maximum output limits.Finally, (16) indicates the limits on output generation of photovoltaic units In many occasions, distribution system operator prefers to avoid several switching actions.In order to prevent unnecessary switching actions, any changes in status of switches must be detected.In (17) and (18), if the status of a switch changes, binary variable BM t, l becomes 1, otherwise, it is 0. Also, ( 19) is considered to ensure that the status of each MS can be changed only once.Similar to the MSs, status of RCSs cannot be changed more than once.These conditions for RCSs are presented in ( 20)- (22), respectively As discussed heretofore, any changes in the status of MSs call for dispatching a crew team.In order to model this feature of MSs, at first, it is supposed that crew teams are located in the depots before the occurrence of the natural hazard.Furthermore, the distance between different switches is assumed to be known for the distribution system operator.The following equations are considered for modelling the crew dispatching to the switch locations In the above, ( 23) and ( 24) represent the arrival time of crew cr to the location of point j.If the crew arrives to the location of point i at time time i cr , it spends time ct to change the status of the switch, and then it spends time ctt i, j to move from point i to point j.Therefore, in case crew cr travels from point i to point j (r i, j cr ), the arrival time is equal to time i cr + ctt i, j + ct.As can be seen, if crew cr does not travel to point j, then its travel time should not affect ( 23) and (24), which is addressed by (25).Furthermore, (26) guarantees that all of the crew teams start their travel from the staging locations.As can be seen, (27) ensures that status of an MS can be changed only if a crew moved to its location, and vice versa.Moreover, (28) indicates that once a crew moves to the location of a switch, it has to leave there after changing status of the switch.
As can be seen, ( 29) implies that at most one crew can move to location of a switch.All the crew teams should be dispatched to the staging locations after accomplishing the actions, which is adhered by (30).In addition, using (31) and (32), the status changing time of switches can be determined.Constraint (33) indicates that a crew can be dispatched from point i only if it is already moved there.Furthermore, (34) implies that none of the crews should be dispatched to RCS locations since the status of the switches can be altered remotely.Finally, (35) indicates that the time required for switching operations should be less than predictability of the event (POE).

Mid-event monitoring
In this study, it is supposed that system operators give up any remedial actions during the event mainly to preserve their safety.The operators just monitor the situation in order to gather real-time data about the operating status of different system components.This monitoring action is necessary for taking the right decisions after the event is finished.

Post-event scheduling
After the event hits the distribution network, the true status of all lines (whether damaged or not) is revealed.In this stage, the event is finished, and the system experiences a new operating condition.
Here, the objective is to minimise energy curtailment caused by the natural hazard.The post-event scheduling is formulated as follows: The objective is selected according to the fact that serving critical loads is the main goal of the system operator in extreme conditions.In (36), ω n is to consider the priority of loads hosted by different buses.The objective is subject to the following technical constraints: Constraints ( 38) and ( 39) indicate the active and reactive power balance at each bus.The term br t, l in (40) stands for availability of the associated line.Constraints ( 41) and ( 42) are considered for power flow limit of each line.The maximum allowable amount of load curtailment at each bus is presented in (43) which indicates that the maximum load curtailment is equal to the hosted load.
In order to maintain the radial structure of the distribution system, this study applies the spanning tree approach wherein each bus except the root bus has either one or zero parent bus.Detailed information about the approach is given in [26].As can be seen, (44) ensures that line l at time t is in the spanning tree (br t, l = 1) if either γ m, n, t = 1 which means bus n is the parent of bus m or γ n, m, t = 1 which means bus m is the parent of bus n.As shown in (45), each bus other than the main grid bus has one or less parent bus.Meanwhile, (46) guarantees that the main grid bus has no parent.
Finally, it is worthwhile to mention that natural disasters can be divided into two groups, namely predictable events and unpredictable ones.For example, hurricane, flood, tropical storm, and blizzard are predictable natural hazards, while it is impossible to predict some natural hazards like an earthquake.According to [1], predictability of an earthquake is from seconds to minutes, while predictability of a hurricane is from 24 to 72 h.Therefore, significance and motivation for the pre-event modelling stage are regarding events like hurricane and flood that can be anticipated in advance.Accordingly, due to unpredictable nature of some natural disasters (e.g.earthquake), system operator does not have the opportunity to implement the pre-event scheduling for these kinds of events and he/she only implements the second and third stages of the proposed model.

Test system under study and main assumptions
The proposed approach is implemented on a 20 kV electric power distribution system.The geographic map and single line diagram of the system are shown in Figs. 3 and 4, respectively.As can be seen, this system includes four distribution feeders which feed 47 distribution transformers.The system hosts industrial, commercial, and residential customers.The feeders are equipped with three normally open switches (installed in lines 48, 49, and 50).It is worthwhile to mention that three of the normally closed switches can be controlled remotely (installed in lines 8, 26, and 43).The feeders consist of Fox conductor with approximate thermal capacity of 10 MVA and feeder parameters (B l and G l ) are taken from [27].Additionally, a photovoltaic unit is connected to bus 20, and three wind turbines are hosted by buses 26, 33, and 37.
In the system, two 5 MW distributed generation units are assumed to be hosted by buses 9 and 43.Also, the minimum and maximum reactive powers of the DGs are −5 and 5 MVAr, respectively.Moreover, the hourly output powers of the photovoltaic units and wind turbines are demonstrated in Figs. 5  and 6, respectively [27].Three 0.35 MWh storage units are hosted by buses 19, 24, and 33.The lower and upper limits for bus voltage magnitudes are considered 0.95 and 1.05 p.u., respectively.
The system load profile is shown in Fig. 7.As can be seen, system peak demand is equal to 35 MW which occurs around 9 PM.It should be noted that, in this study, geographical information of the power distribution system buses is assumed to be available for the system operator (see Fig. 3).Therefore, the distance and Fig. 3 Geographical map of the distribution system [27] Fig. 4 Single-line diagram of the distribution system Fig. 5 Output power of photovoltaic units [27] IET Smart Grid, 2019, Vol. 2 Iss. 4, pp.549-556 This is an open access article published by the IET under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/)travelling time between different points of the system can be determined using the available software packages and applications like Google map.In this paper, a typical day is considered and the software packages are used to determine the required time for travelling between different points in the system.It is assumed that the system operator fixes the locations of crew teams before disturbance begins and these locations are used to make further decisions for dispatching crew teams.Regarding the status of the crew teams, it is assumed that the communication between the operator and crew teams is stable during the event.This is not an unrealistic assumption since there is usually a back-up communication system for emergency conditions.For example, system operators can use radio communication systems as an emergency communication system during disasters instead of public communication systems.Since the travelling times are based on normal conditions, they are multiplied by traffic congestion factor (TCF) to consider potential traffic issues during emergency conditions, where TCF is defined as emergency condition travelling time over the normal condition travelling time.It is worthwhile to note that the value of TCF is set to two in this study.Besides, it is assumed that two crew teams are available for the system operator.It is assumed that the approaching disturbance hits the system at 1 PM.Moreover, it is assumed that the failure probability of system lines is determined according to the predicted disturbance characteristics.

Simulation results
To evaluate effectiveness of the proposed method, two different cases are considered as follows: Case I: In this case, no pre-event scheduling is conducted and all remedial actions are taken after the approaching disturbance hits the network.To reach this goal, in this case, it is assumed that POE is equal to zero, which is equivalent to the post-event procedures and methodologies available in the literature.
Case II: This case simulates the proposed method where both preevent and post-event actions are taken to minimise negative impacts of the disturbance.In this case, it is assumed that POE is equal to 30 min.Therefore, in this case, the system operator is capable of scheduling the distribution system in advance of the approaching calamity in order to use the available redundant paths with lower failure probability and lower power flow.These preevent actions may lead to a better configuration of the system to cope with the natural hazard.
The two cases are simulated and the associated results are provided.The simulations are conducted on a PC with Intel Core i5 CPU @3.4 GHz and 8 GB RAM.In the simulations, optimisation problems are in GAMS environment.
The first case is simulated and the associated results are given in Table 1.It is assumed that vulnerable lines with failure probability larger than 5% are damaged after natural disaster strikes.As can be seen, in the presence of two crew teams, the optimal sequence of the actions taken by the crew teams are d p, SW1, d p and d p, SW2, SW3, d p , respectively.At first, crew number one is dispatched from the depot to close SW1, which takes 29 min with considering value of two for TCF (i.e.24 min for arrival plus 5 min for switching).Simultaneously, the second crew travels to close SW2.This action takes 35 min.Then, the same crew is dispatched to close SW3, which takes 60 min.In this case, scheduling of crew teams in the distribution system after the occurrence of the event is shown in Fig. 8.As illustrated in Fig. 9, the amount of load supplied by the distribution system falls to 57.3% after the event hits, which is restored to 89.8% an hour after ending of the event via post-event scheduling.
In the second case, the system operator is able to foresee the approaching disturbance half an hour in advance.Moreover, crew teams are dispatched to the staging locations after receiving the first hints about the potential approaching disaster.Based on these forecasts, the pre-event optimisation problem is solved and the available crew teams are dispatched to take the switching actions listed in Table 2.As can be seen in the table, the optimal sequence of crew dispatching to the location of MSs is defined as sl, L30, SW2, sl and sl, L18, SW3, sl , where sl is the staging location.Therefore, scheduling of the crew teams in advance of the event in the distribution system is illustrated in Fig. 10.
In the post-event stage, system operators run the third stage optimisation problem with the aim of minimising energy curtailment caused by the natural disaster.Therefore, the first crew team is dispatched to the location of SW1 with a total time of 15 min after the event (i.e. 10 min for arrival plus 5 min for switching).Scheduling of the crew teams in the post-event stage is shown in Fig. 11.With taking this action into account, total Fig. 6 Output power of wind turbines [27] Fig. 7 Load profile of the distribution system [27]   supplied load of the system rises to 89.8%.Needless to mention, these actions (i.e.post-event actions) take more time as the traffic congestion gets acuter after a big disaster hits the system.It is worthwhile to mention that the system operator could not conduct any pre-event scheduling if POE is less than the required time for accomplishing the first action.Fig. 9 compares the performance of the two cases.As can be seen, in the second case, the percentage of supplied load drops to 78% when the lines are broken.In addition, using the proposed approach, supplied load level reaches to 89.8% within 15 min.In addition, as can be seen in Fig. 9 within one hour after occurrence of the event, 8.44 MWh of the load is curtailed in case I which is equal to $101,280 damage cost with considering 12,000 $/MWh for the value of lost load [28].While, only 3.65 MWh of the total load is curtailed in case II, which is equal to the total damage cost of $43,800.Therefore, the proposed model leads to a lower damage cost compared to existing post-event methodologies.Note that, the duration and severity of the interruptions depend on the system as well as the event severity.In the studied system, mainly because of its flexibility and small size and partly due to the severity of the event, 90% of system load is restored within an hour.It is clear if more system lines are damaged or the network is larger (distances between points are longer), the restoration could take longer.In the simulated system, the required time for travelling between the MSs in the system is short.Also, the restoration process of distribution systems after a major disaster is time consuming mainly due to the repair process of damaged lines, while in this paper, because of flexibility of the network, significant portion of the load is restored via DGs and redundant paths.In addition, it is worthwhile to note that about 10% of the system load cannot be restored until some of the damaged lines are repaired/replaced which may take much longer.
In addition, with taking the load criticality into account more critical load is curtailed in case I than in case II.According to the comparisons, the proposed approach significantly outperforms the existing method which only takes post-event actions.

Sensitivity analyses
A sensitivity analysis is performed to investigate the impact of events predictability on conservation level of results and required time for taking post-event actions with two available crews.As shown in Table 3, POE has a significant effect on post-event action's rapidity.This is mainly due to the fact that taking associated actions in post-event stage consumes more time because of traffic issues.To obtain the results provided in Table 3, the preevent scheduling problem is simulated for different POE values.As a result, pre-event configuration of the system is achieved for different scenarios with different POE values.Since the system configurations are different for different scenarios, restoring the system (as much as possible) in the post-event stage takes different time durations in different scenarios.To calculate these time durations, the post-event scheduling problem is solved for the scenarios considering the associated pre-event configurations as the initial configuration.Then, considering the actions taken in different scenarios, the required time for restoring the system (as much as possible) in the post-event stage is calculated.For example, in the scenario with POE equals to 30 min, the system operator dispatches the crew number one to close SW1 (switch of line 48) which takes 15 min.This means that the required time for post-event actions is equal to 15 min in the scenario where POE is equal to 30 min.Also, to demonstrate the impact of available crew teams and possible traffic issues after event hits the system on total required time for taking the post-event actions, a sensitivity analysis is conducted and associated results are shown in Fig. 12.As can be seen, the number of available crew teams, as well as the TCF, have a notable impact on the total time required to take the associated actions even in a small region where the switches are close to each other.

Conclusion
In this paper, in order to enhance resilience of distribution systems against high-impact low-probability events, a new three-stage scheduling model has been proposed.In the first stage, failure probability of the lines is determined based on approaching event's severity and components' fragility curves.Then, an optimisation problem is run to obtain the optimal pre-disturbance network configuration such that system loads are served through less vulnerable paths.At the second stage, the system operator monitors    the operating condition of different elements.Then, an optimisation model is run to optimise post-event remedial operations, if necessary.The proposed model is applied to a typical distribution network.According to the results, applying pre-event actions can significantly reduce load curtailments caused by disasters.Moreover, various sensitivity analyses are conducted on key affecting parameters.The results also confirm that in facing natural disasters, the more system operators have time to apply the preevent scheduling, the more the system will be resilient against the disturbance.

Fig. 1 Fig. 2
Fig. 1 Main stages of the proposed method

Fig. 8 Fig. 9
Fig. 8 Scheduling of crew teams in the post-event stage in case I

Fig. 10
Fig. 10 Scheduling of crew teams in the pre-event stage in case II

Fig. 12
Fig. 12 Total time to take post-event operations with different TCFs and different numbers of crew teams

Table 1
Post-event switching operations in case I No. cr.

Table 2
Pre-event switching operations in case II No. cr.Disp.crew Sw.No. Status Total time, min

Table 3
Required time for post-event actions for different POE values predictability, min 55 30 20 10 required time for post-event actions, min 0 15 30 56