stochastic pre-event preparation for enhancing resilience
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Contents lists available at ScienceDirect
Renewable and Sustainable Energy Reviews
journal homepage: www.elsevier.com/locate/rser
Stochastic pre-event preparation for enhancing resilience of distributionsystemsQianzhi Zhang a,β, Zhaoyu Wang a, Shanshan Ma b, Anmar Arif c
a Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USAb School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 85287, USAc Department of Electrical Engineering, King Saud University, Riyadh 11451, Saudi Arabia
A R T I C L E I N F O
Keywords:Extreme weather eventsPre-event preparationPV penetrationResilienceResource allocationTwo-stage stochastic model
A B S T R A C T
Extreme weather events are the common causes for power supply interruptions and power outages in electricaldistribution systems. Improving the distribution system and enhancing its resilience is becoming crucial due tothe increased frequency of extreme weather events. Preparation and allocation of multiple flexible resources,such as mobile resources, fuel resources, and labor resources before extreme weather events can mitigate theeffects of extreme weather events and enhance the resilience of power distribution systems. In this paper, atwo-stage stochastic mixed-integer linear programming (SMILP) is proposed to optimize the preparation andresource allocation process for upcoming extreme weather events, which leads to faster and more efficientpost-event restoration. The objective of the proposed two-stage SMILP is to maximize the served load andminimize the operating cost of flexible resources. The first stage in the optimization problem selects theamounts and locations of different resources. The second stage considers the operational constraints of thedistribution system and repair crew scheduling constraints. The proposed stochastic pre-event preparationmodel is solved by a scenario decomposition method, Progressive Hedging (PH), to ease the computationalcomplexity introduced by a large number of scenarios. Furthermore, to show the impact of solar photovoltaic(PV) generation on system resilience, three types of PV systems are considered during a power outage and theresilience improvements with different PV penetration levels are compared. Numerical results from simulationson a large-scale (more than 10,000 nodes) distribution feeder have been used to validate the effectiveness andscalability of the proposed method.
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1. Introduction
In recent years, the relationship between climate change, extremeweather events, and power outages have become the focus of discussionworldwide [1,2]. The aging infrastructure of the electric grid combinedwith the increase in severe weather events have highlighted the harshreality of how vulnerable the distribution grid is. For example, hightemperatures from heatwaves will limit the amount of energy that canbe transferred [3], lightning strikes cause faults on the lines [4], andthe high winds from storms may damage overhead lines [5]. In theU.S., extreme weather events have caused 50% to 60% of the powerinterruptions [6] and $20 to $55 billion annual economic losses [7].To mitigate the impacts of extreme weather events on electric infras-tructures and power grids, extensive efforts have been devoted towardproposing the concept of resilience. In [8], resilience was defined as aproperty of systems representing their response to and recovery fromlow probability and high impact events. The measurements of system
β Corresponding author.E-mail addresses: [email protected] (Q. Zhang), [email protected] (Z. Wang), [email protected] (S. Ma), [email protected] (A. Arif).
resilience are disciplined into ecological resilience [9], psychologicalresilience [10], risk management [11], and energy security [12].
About 90% of weather-related power interruptions and outagesare led by failures in distribution systems [13]. Various resilience-enhancing strategies have been studied in distribution systems [14],such as the long-term planning, the pre-event preparation, and thepost-event restoration. The long-term planning provides utility compa-nies the actionable resilience-enhanced methods to upgrade infrastruc-tures in the long-term [15]. For example, the optimal line hardeningstrategies against extreme weather-related hazards are developed tophysically improve electric infrastructure and enhance the long-termresilience of the distribution system in [16β19]. The post-event restora-tion is used by utility companies to prioritize service restoration efforts,schedule repair crews and manage network reconfiguration after theextreme weather events [20]. For example, the dynamic formation ofmicrogrids (MGs) and optimal coordination between multiple MGs are
https://doi.org/10.1016/j.rser.2021.111636Received 23 March 2021; Received in revised form 19 July 2021; Accepted 27 Aug
ust 2021Q. Zhang et al.
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Notations
List of abbreviations
CI Confidence intervalDERs Distributed energy resourcesDGs Distributed generatorsEF Extensive formESS Energy storage systemMEGs Mobile emergency generatorsMESs Mobile storage devicesMRP Multiple replication procedurePH Progressive hedgingPV Photovoltaic generationSOC State of chargeSMILP Stochastic mixed-integer linear program-
ming
Indexes
π Index of conductor between polesπ, π Index of busππ Index of line between bus π and bus ππ Index of conductor or lineπ Index of network loopπ Index of poleπ Index of scenarioπ‘ Index of time instant
Sets
πΊB Set of line switchesπΊCN Set of candidate buses for MEGs and MESsπΊDL Set of damaged linesπΊEG Set of buses that have fuel-based emergency
generatorsπΊES Set of buses with ESSsπΊESC Set of buses with all types of storage unitsπΊG Set of generatorsπΊK Set of linesπΊloop Set of network loopsπΊN Set of busesπΊR Set of network regionsπΊPV Set of PV systemsπΊG
PV Set of grid-following PV systemsπΊH
PV Set of hybrid on-grid/off-grid PV systemsπΊC
PV Set of grid-forming PV systems
Parameters
π, ππ Coefficients associated with the compactfirst stage variable π₯ and compact secondstage variable π¦π
πΆF Unit cost of fuel consumption of generators(LβkWh)
πΆSW Unit cost of line switches ($)πΆDπ Unit cost of load shedding ($/kWh)
πpπ,π,π‘ Active power demand of bus π, phase π andtime π‘
πΈCapπ Maximum capacity of ESSs of bus π
considered to restore the critical loads and services during power out-
ages in [21β23]. In this paper, we focus on the pre-event preparation,
πΈSOC,maxπ , πΈSOC,min
π Maximum and minimum permissible rangeof SOC of bus π
πCh,maxπ , πDis,max
π Maximum charging and discharging powersof ESS of bus π
πK,maxπ , πK,max
π Active and reactive power flow limitsπG,maxπ , πG,max
π Active/reactive power output limits of gen-erator
π PVπ,π,π‘,π Active power output of PV systems of bus π,
phase π, time π‘ and scenario π π rateπ Rate capacity of PV systems of bus π
π πππ,ππ (π€(π‘)) Failure probability of the overhead line ππwith wind speed π€ at time π‘
π πππ,ππ,π(π€(π‘)) Failure probability of the pole π at line ππwith wind speed π€ at time π‘
π πππ,π (π€(π‘)) Failure probability of conductor π betweentwo poles with wind speed π€ at time π‘
π πππ€,π (π€(π‘)) Direct wind-induced failure probability ofconductor π with wind speed π€ at time π‘
π πππ‘π,π (π€(π‘)) Fallen tree-induced failure probability ofconductor π with wind speed π€ at time π‘
π ππ’,π Probability that conductor π is undergroundππ(π ) Probability of occurrence for scenario π πππ,π Phases π of line between bus π and bus ππESS,max
π Maximum limit of reactive power output ofESS
π€(π‘) Wind speed at time π‘ππ Median capacity of conductorπMEG Number of available MEGsπMES Number of available MESsπMU
π Number of mobile unitsπFuel
π Amount of available fuelπCrew Total number of crews.πCrew,max
π , πCrew,minπ Maximum and minimum number of avail-
able repair crews of region πππππππ’ππ‘ππ Number of conductor wires between two
adjacent polesπππππ Number of distribution poles supporting
lineοΏ½ΜοΏ½ππ , οΏ½ΜοΏ½ππ Unbalanced three-phase resistance matrix
and reactance matrix of line πππF Rate between fuel consumption and power
output of generators (L/kWh)πPVπ PV capacity of bus π
πΌπ Solar irradianceπminπ , πmax
π Maximum and minimum limits of squaredvoltage of bus π
πΌ Average tree-induced damage probability ofoverhead conductor
ππ Logarithmic standard deviation of intensitymeasurement
πCh, πDis ESS charging and discharging efficiencies
which helps utility companies to prepare resources in advance and mit-igate the upcoming extreme weather events. The pre-event preparationcan not only avoid high investment cost in long-term planning, but alsoefficiently reduce the outage duration in post-even restoration.
There are existing studies that have investigated pre-event prepara-tion and resource allocation problems for the resilience enhancementof electric distribution systems. In [24β26], pre-event resource manage-
ment in MGs and pre-event operation strategies in distribution systems 10Q. Zhang et al.
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π Iteration number of PHπ Penalty factor of PHπ Termination threshold of PH
Continuous Variables
πΈSOCπ,π‘,π SOC of ESS of bus π, time π‘ and scenario π
πΉπ,π Total fuel consumption of generatorsπKππ,π,π‘,π , π
Kππ,π,π‘,π Active/reactive power flows of line ππ,
phase π, time π‘ and scenario π πGπ,π,π‘,π , π
Gπ,π,π‘,π Active/reactive power outputs of fuel-based
generator of bus π, phase π, time π‘ andscenario π
πChπ,π,π‘,π , π
Disπ,π,π‘,π Active charging/discharging power output
of ESS of bus π, phase π, time π‘ of scenarioπ
πESSπ,π,π‘,π Reactive power output of ESS of bus π, phase
π, time π‘ and scenario π πPV
π,π,π‘,π Reactive power output of PV of bus π, phaseπ, time π‘ and scenario π
πFuelπ Amount of fuel allocated to the generator ofbus π
πCrewπ Number of repair crews of region πππ,π,π‘,π Square of voltage magnitude of bus π, phase
π, time π‘ and scenario π π£S Virtual sourceπ£fπ Virtual flow of line ππ₯, π¦π Compact first stage and second stage vari-
ablesοΏ½ΜοΏ½ Expected value of first stage variable
Discrete Variables
βπ,π‘,π Binary variable indicating if ESS is charg-ing/discharging (1) or not (0) of bus π,phase π, time π‘ and scenario π
πMEGπ , πMES
π Binary variable indicating if an MEG or MESis allocated (1) or not allocated (0) to bus π
π’π,π‘,π Binary variable indicating if line π is ener-gized (1) or not (0) of time π‘ and scenarioπ
π¦π,π‘,π Binary variable indicating if load is restored(1) or not (0) of bus π, phase π, time π‘ andscenario π
π§π,π‘,π Binary variable indicating if line π is beingrepaired (1) or not (0) of time π‘ and scenarioπ
πΎππ,π‘,π Binary variable indicating if switch is closed(1) or not (2) of line ππ, phase π, time π‘ andscenario π
ππ,π‘,π Binary variable indicating if bus π is ener-gized (1) or not (0) of time π‘ and scenarioπ
are considered to enhance system resilience during extreme events.In [27], the position and number of depots are determined, and theavailable resources are managed at the pre-event stage. In [20], repaircrews are pre-allocated to depots and integrated with the restora-tion process for enhancing the response after a disaster. A two-stagestochastic model is developed in [28] to determine staging locationsand allocate repair crews for disaster preparation while consideringdistribution system operation and crew routing constraints. In [29],
the authors developed a stochastic model for optimizing pre-eventoperation actions. The study optimized the topology of the network andthe position of crews for upcoming disturbances. In [30] and [31], atwo-stage framework is developed to position mobile emergency gener-ators (MEGs) for pre- and post-disasters. Mobile energy storage devices(MESs) are investigated in [32] and [33] for the resilience enhancementof power distribution systems. However, there are limitations in theabove studies on pre-event preparation and resource allocation. Theselimitations are described in the following:
(1) Pre-event allocation of various flexible resources: In practice, pre-vent preparation includes allocating various flexible resources, suchs MEGs, MESs, fuel resources for diesel generators, and repair crews.he optimal allocation of those flexible resources can help utilities tochieve faster and more efficient post-event power restoration. How-ver, previous studies mainly focused on allocating specific flexibleesources, rather than formulating a complete optimization problem tore-allocate various flexible resources together.
(2) Impacts of solar PV power on system resilience: Due to intermit-ent characteristic of traditional distributed energy resources (DERs),uch as solar power, PV systems are not considered as a reliableesilient solution [34]. However, the distributed nature of PV poweran contribute to a more resilient power system [35]. In practice, PVystems can be coupled with energy storage technologies to enable grid-upporting capability [36], continuous operation during outages [37,8], and economic operation [39,40]. Different types of PV systems andhe impacts of different PV penetration levels on system resilience aregnored in most existing research works.
(3) Scalability of the solution algorithm: On one side, the stochasticre-event preparation model may suffer from computational ineffi-iency due to a large number of scenarios; on the other side, a limitedumber of scenarios may influence the stability and quality of theolutions. Therefore, the trade-off between computation time and solu-ion accuracy needs to be studied for stochastic pre-event preparationethods. In addition, a large-scale system is needed to verify the
calability of solution algorithms.To address these challenges, a two-stage stochastic mixed-integer
inear program (SMILP) is proposed for pre-event preparation with there-allocation of mobile resources, fuel resources and labor resources.urthermore, the proposed pre-event preparation model considers dif-erent types of PV systems and facilitates the benefits of leveraging highV penetration for improving the resilience of distribution grids. In thisaper, resilience improvement is quantified by the increased servedoad and reduced outage duration. To deal with the massive compu-ation burden, the proposed two-stage stochastic pre-event preparationroblem is solved by a scenario decomposition method, Progressiveedging (PH) [41], while maintaining the accuracy and stability of
he solution [42]. Also, the quality of the solution is validated by theultiple replication procedure (MRP) [43]. The main contribution of
his paper is three-folded:
β’ A two-stage SMILP model is proposed for pre-event preparationfor upcoming extreme weather events, where the first stage allo-cates MEGs, MESs, fuel, and repair crews. The second stage con-siders distribution system operation and repair crew schedulingconstraints.
β’ The proposed pre-event preparation model considers three typesof PV systems during a power outage, including grid-followingPV system, hybrid on-grid/off-grid PV system and grid-formingPV system. The improvements of resilience and the reductionof outage duration with different PV penetration levels are alsopresented.
β’ The proposed solution algorithm is tested through a solutionvalidation method to show its quality. In addition, a large-scalesystem, consisting of more than 10,000 nodes, is used to verifythe scalability of the proposed pre-event preparation model.
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The remainder of the paper is organized as follows: Section 2describes the proposed two-stage SMILP for pre-event preparation andresource allocation. Section 3 presents the PH solution algorithm, con-vergence analysis and solution validation. Case study and results dis-cussion are given in Section 4. Conclusions are provided in Section 5.
2. Two-stage stochastic pre-event preparation model
The general framework of the proposed two-stage stochastic pre-event preparation model is shown in Fig. 1. Damage scenarios ofextreme weather events are generated based on the following infor-mation: (1) identification of extreme weather events, such as flood,hurricane and winter storm; (2) extreme weather event data and metric;(3) fragility model of test systems, which describes the behavior ofcomponents under extreme weather events; (4) damage status of com-ponents in test systems subject to specific extreme weather events. Toapproximate the impact of extreme weather events to grid infrastruc-tures, damage scenarios can be generated by mapping the weather dataset to the failure probability of grid infrastructures. The Monte Carlosampling technique can be used to generate a manageable numberof scenarios. Adopted from [44], for wind speed π€(π‘), the relatedfailure probability ππππ,ππ (π€(π‘)) of overhead line ππ can be formulatedas follows:
ππππ,ππ (π€(π‘)) = 1 βπππππβ
π=1
(
1 β ππππ,ππ,π(π€(π‘)))
ππππππ’ππ‘ππβ
π=1
(
1 β ππππ,π (π€(π‘)))
(1)
where ππππ,ππ,π(π€(π‘)) and ππππ,π (π€(π‘)) are the failure probability of poleπ at line ππ and the failure probability of conductor π between twopoles, respectively. πππππ represents the number of distribution polessupporting line ππ and ππππππ’ππ‘ππ represents the number of conductorwires between two adjacent poles at line ππ, respectively. In Eqs. (2)and (3), ππππ,ππ,π(π€(π‘)) and ππππ,π (π€(π‘)) can be expressed as follows:
ππππ,ππ,π(π€(π‘)) = π·[
ln(π€(π‘)βππ
ππ )]
(2)
πππ,π (π€(π‘)) = (1 β πππ’,π ) max(
ππππ€,π (π€(π‘)), πΌπ πππ‘π,π (π€(π‘)))
(3)
where π· is the operator of the log-normal cumulative distributionfunction (CDF). ππ and ππ are the median capacity and the logarithmicstandard deviation of intensity measurement, respectively; ππππ€,π (π€(π‘))represents the direct wind-induced failure probability of conductor πand ππππ‘π,π (π€(π‘)) represents the fallen tree-induced failure probability ofconductor π. πππ’,π is the probability that conductor π is underground,which is more invulnerable to extreme weather events. πΌ representsthe mean probability of tree-induced damage for overhead conductors.More details of weather forecasting methodologies, line fragility modelsand scenario generation can be found in [45].
As shown in Fig. 1, the proposed SMILP pre-event preparationmodel has two stages: (i) Flexible resources are pre-allocated for up-coming extreme weather events in the first stage, including the optimaldecisions of pre-position and number of MEGs, MESs and repair crewsto depots, and allocation of available fuel to generators. (ii) The sec-ond stage determines the optimal hourly operation of the distributionsystems and assigns repair crews to the damaged components afterthe extreme weather events. Constraints in the second stage includeunbalanced optimal power flow constraints, network reconfigurationand isolation constraints, and repair crew scheduling constraints.
2.1. Objective function
The objective function (4) is set to minimize operating cost andmaximize the served loads. There are three unit cost coefficients inthe objective, unit cost of fuel consumption πΆF (LβkWh), unit cost of
switching operation πΆSW ($), and unit cost of load shedding πΆDπ at bus
π ($/kWh). The objective is formulated as follows:
minβ
βπ ππ(π )
(
πΆFπFβ
βπ‘
β
βπ
β
βππGπ,π,π‘,π + πΆSW
β
βπ‘
β
βπβπΊSW
πΎππ,π‘,π
+β
βπ‘
β
βπ
β
βππΆDπ (1 β π¦π,π‘,π )π
pπ,π,π‘
)
(4)
where ππ(π ) is the probability of occurrence for scenario π . Based onthe total number of scenarios ππ , ππ(π ) can be calculated as 1β|ππ |. πFis the rate between fuel consumption and energy output of generators.The unit of πF is πΏβππ β, which represents the fuel consumption in πΏper energy generation in ππ β. πG
π,π,π‘,π is the active power output forfuel-based generator at bus π, phase π, time π‘, and scenario π . Binaryvariable πΎππ,π‘,π represents the status of each switch, if a switch on line ππis operated at time π‘, and scenario π , then πΎππ,π‘,π = 1. The binary variableπ¦π,π‘,π represents the status of load at bus π, time π‘, and scenario π . If thedemand πpπ,π,π‘ is served, then π¦π,π‘,π = 1.
2.2. First stage constraints
The first stage constraints revolve around pre-allocating four impor-tant resources that will be utilized after an extreme event: (i) MEGs, (ii)MESs, (iii) fuel and (iv) repair crews.
2.2.1. Mobile resources allocation constraintsMobile resources can be used to restore energy for isolated areas
that are not damaged, and to restore critical loads. In addition, fuelmanagement is important after an extreme event to operate emergencygenerators. Distributing fuel after an extreme event may be difficultdue to road conditions. As for repair crews, pre-assigning them todifferent locations provides a faster and more organized response. Theconstraints for allocating the mobile resources are modeled as follows:
β
βπβπΊCN
πMEGπ = πMEG (5)
β
βπβπΊCN
πMESπ = πMES (6)
πMEGπ + πMES
π β€ πMUπ ,βπ β πΊCN (7)
where binary variables πMEGπ and πMES
π equal 1 if a MEG or MES areallocated to bus π, respectively. The set πΊCN represents the set ofcandidate buses for MEGs and MESs. Constraints (5) and (6) indicatesthat the number of installed MEGs and MESs are equal to the numberof available devices (πMEG and πMES). In this work, it assumes thateach bus can only have a limited number of mobile units πMU
π , whichis enforced by (7).
2.2.2. Fuel resources allocation constraintsDefine the set πΊG = πΊEG βͺ πΊCN, where πΊEG is the set of buses
that have fuel-based emergency generators. The fuel allocated to πΊGmust be limited to the available amount of fuel. The fuel allocationconstraints are presented as follows:β
βπβπΊπΊ
πFuelπ β€ πFuel (8)
πΉGπ β€ πFuelπ β€ πΉmax
π ,βπ β πΊG (9)
Constraint (8) limits the total amount of allocated fuel to the avail-able amount of fuel (πFuel), where πFuelπ is the amount of fuel allocatedto the generator at bus π. In this work, it assumes that not all theavailable fuel needs to be allocated. Constraint (9) limits the amountof fuel on each site, where πΉG
π is the amount of fuel already present forthe generator at bus π, and πΉmax represents the maximum capacity of
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Fig. 1. The proposed two-stage stochastic pre-event preparation model.
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fuel at bus π. Note that the aim of fuel allocation is to decide how muchfuel should be allocated to the generators, which have defined locationsin the system. Therefore, the logistic process of transferring the fuel tothe generators is not considered in this paper, as this problem can besolved as a separate problem.
2.2.3. Repair crew allocation constraintsTo allocate the repair crews, our model divides the network into
different regions πΊR. Each region will be assigned with different crews,who will conduct the repairs in that region. Note that the buses in asingle region should be relatively close to each other. These regionsshould be determined based on the physical distances between thebuses. Then the crews are allocated to the regions, where the crewswould be stationed at a depot. Therefore, the distance is not explicitlyconsidered in the mathematical model, but it is considered in thepreprocessing step of determining the regions.
Constraint (10) states that the total crews deployed to all regionsis equal to the number of available crews. This constraint can berelaxed by replacing the equality with an inequality if some crews arerequired on standby. This work assumes that all available crews willbe deployed. Constraint (11) sets a minimum and maximum number ofcrews that can be stationed in each individual region.β
βπβπΊR
πCrewπ = πCrew (10)
πCrew,minπ β€ πCrewπ β€ πCrew,max
π ,βπ β πΊR (11)
where πCrewπ is the number of repair crews in region π and πCrew isthe total number of crews. The number of repair crews is limited ineach region, using πCrew,min
π and πCrew,maxπ , depending on the size and
capacity of the staging locations.After allocating the fuel in the first stage, each generator can be
operated in the second stage based on how much fuel is available.Similarly, once the pre-position decisions of mobile resources andrepair crews are obtained in the first stage, the second stage can decidethe mobile resource operation and repair schedule.
2.3. Second stage constraints
In the second stage of the proposed pre-event preparation model,the constraints of PV systems and repair crew dispatch are mainly
(discussed. The model also considers unbalanced power flow constraints,voltage constraints, and network reconfiguration constraints [43,46].
2.3.1. PV system constraintsTo thoroughly investigate the impact of PV systems on system
esilience, three types of PV systems are considered with differentperation modes in the second stage [43], πΊPV = πΊG
PV βͺ πΊHPV βͺ πΊC
PV.The main differences between those three types of PV systems are theirdifferent behaviors during an outage: (i) Type 1: on-grid PV with grid-following operation mode (πΊG
PV), where the PV will be switched off anddisconnected during an outage. (ii) Type 2: hybrid on-grid/off-grid PV+ energy storage system (ESS) (πΊH
PV), where the PV system operateson-grid in normal condition or off-grid during an outage (serves localload only). (iii) Type 3: grid-forming PV + ESS with grid-formingcapability (πΊC
PV), this system can restore part of the network thatis not damaged if the fault is isolated. There are several benefits ofconsidering different types of PV systems during a power outage. Forexample, this kind of model is more like a real-world application withmultiple PV systems. In addition, the PV systems are mostly consideredas power supply resources in previous research works, while the grid-forming and black-start capability of PV systems during outages shallalso be explored and discussed. The output power of the PV systems isdetermined using the following equations:
0 β€ π PVπ,π,π‘,π β€
πΌππ,π‘,π 1000 Wβm2
π rateπ ,βπ β πΊPVβπΊG
PV, π, π‘, π (12)
0 β€ π PVπ,π,π‘,π β€ ππ,π‘,π
πΌππ,π‘,π 1000 Wβm2
π rateπ ,βπ β πΊG
PV, π, π‘, π (13)
(π PVπ,π,π‘,π )
2 + (πPVπ,π,π‘,π )
2 β€ (πPVπ )2,βπ β πΊPVβπΊG
PV, π, π‘, π (14)
(π PVπ,π,π‘,π )
2 + (πPVπ,π,π‘,π )
2 β€ ππ,π‘,π (πPVπ )2,βπ β πΊG
PV, π, π‘, π (15)
The PV active power output π PVπ,π,π‘,π depends on the solar cell rating
apacity π rate and the solar irradiance πΌππ,π‘,π [47]. The active powerutputs of Type 2 πΊH
PV and Type 3 πΊCPV PVs can be determined in (12),
hile the active power outputs of Type 1 πΊGPV PVs is calculated in
13). The binary variable π = 0 if bus π is not energized at time
π,π‘,π 70Q. Zhang et al.
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π‘ and scenario π . Using advanced PV smart inverters [48], the PVscan provide reactive power support πPV
π,π,π‘,π , which is constrained bythe capacity πPV
π in (14) and (15). During an outage, on-grid PVs aredisconnected and the on-site load is not served by the PVs, therefore,constraints (13) and (15) are multiplied by ππ,π‘,π . PV systems of typesπΊC
PV and πΊHPV can disconnect from the grid and serve the on-site load.
An example network with a damaged line is given in Fig. 2, wherethe network is divided into three islands due to the damaged line. Thegrid-forming sources in πΊC
PV βͺπΊG has the black start capability and canrestore the network. While PV system in types πΊG
PV or πΊHPV can connect
to the grid only after the PV bus is energized. Island A has a grid-forming generator, therefore, a microgrid is created and the PV systemcan participate. Island B must be isolated because of the damaged line.Island C does not have any grid-forming generators; hence, it will notbe active and the grid-tied PV will be disconnected.
To determine the connection status of the PV systems, a virtual net-work is designed in parallel to the distribution network. The examplenetwork shown in Fig. 2 is transformed into a virtual network shown inFig. 3. To identify if an island network can be energized and restoredby grid-forming sources πΊC
PVβͺπΊG, a virtual network is built with virtualsources, virtual flows, and virtual loads. Each grid-forming generator isreplaced by a virtual source with infinite capacity. Other power sourceswithout grid-forming capability (e.g., grid-tied PVs) are removed. Thevirtual loads with magnitude of 1 replace the actual loads. The virtualnetwork scheme is modeled using constraints (16)β(20).
β
βπβπΊCPVβͺπΊG
π£Sπ,π‘,π +β
βπβπΊK (.,π)π£fπ,π‘,π = ππ,π‘,π +
β
βπβπΊK (π,.)π£fπ,π‘,π ,βπ, π‘, π (16)
β (π’π,π‘,π )π β€ π£fπ,π‘,π β€ (π’π,π‘,π )π,βπ β πΊK , π‘, π (17)
0 β€ π£Sπ,π‘,π β€ (πMEGπ + πMES
π )π,βπ β πΊCN, π‘, π (18)
ππ,π‘,π β₯ π¦π,π‘,π ,βπ β πΊNβ{πΊCPV βͺπΊH
PV βͺπΊG}, π‘, π (19)
ππ,π‘,π + πMEGπ + πMES
π β₯ π¦π,π‘,π ,βπ β πΊCN, π‘, π (20)
A power balance equation is added for each virtual bus, whichmeans that if the virtual load at a bus is served, then that bus is ener-gized. Therefore, for islands without grid-forming generators, all buseswill be de-energized as the virtual loads in the island cannot be served.Constraint (16) is the node balance constraint for the virtual network.Virtual source π£S is connected to buses with power sources that have thecapability to restore the system. The variable π£fπ represents the virtualflow on line π and each bus is given a load of 1 that is multiplied by ππ.Therefore, ππ = 1 (bus π is energized) if the virtual load can be servedby a virtual source and 0 (bus π is de-energized) otherwise. The virtualflow is limited by (17). The limits are multiplied by the status of theline (π’π,π‘,π ) so that the virtual flow is 0 if a line is disconnected. Thevirtual source can be used only if a generator is installed, as enforcedby (18). Define πΊN as the set of all buses. If bus π is de-energized,then the load must be shed (19), unless bus π has a local power sourcewith disconnect switch. Constraint (20) is similar to (19) but with thepresence of mobile sources.
2.3.2. Repair crews constraintsThe second stage of the proposed pre-event model assigns repair
crews to damaged components that are in the area at where the crewsare positioned. Note that the travel time is neglected in this study, asthe travel distances between components in the same area is assumedto be small. An example for crew assignment is given in Fig. 4, wheretwo working areas are assigned for the crews. In this example, fourdamaged lines in Area 1 will be repaired by crews 1β3, while crews 4
and 5 are responsible for the two damaged lines in Area 2. The repaircrews constraints can be presented as follows:
β
βπβπΊDL(s)
π§π,π‘,π β€ πCrewπ ,βπ, π‘, π (21)
β
βπ‘π§π,π‘,π β€ π π
π,π ,βπ β πΊDL(s), π (22)
1π ππ,π
π‘β1β
π=1π§π,π,π β 1 + π β€ π’π,π‘,π β€
1π ππ,π
π‘β1β
π=1π§π,π,π ,βπ β πΊDL(s), π‘, π (23)
where π§π,π‘,π is a binary variable, π§π,π‘,π = 1 means that line π is beingrepaired at time π‘ on scenario π , and πΊDL(s) is the set of damaged lineson scenario π . Constraint (21) limits the number of repairs being con-ucted in each area according to the number of crews πCrewπ available.
Constraint (22) defines the repair time for each damaged line. The linestatus π’π,π‘,π equals 0 until the repair process is conducted for π π
π,π timeeriods. Based on constraint (23), let π π
π,π = 3, π§π,π‘,π = {0, 0, 1, 1, 1, 0, 0},hen π’π,π‘,π = {0, 0, 0, 0, 0, 1, 1}. For example, when π‘ = 6 and π = 0.001,hen constraint (23) becomes 0.668 β€ π’π,6,π β€ 1, therefore, π’π,6,π = 1.
.3.3. Network operational constraintsThe next set of constraints are related to the operation of distri-
ution systems, including unbalanced power flow equations, radialityonstraints, fuel consumption, and energy storage constraints. Thenbalanced distribution system constraints are given below:β
πβπΊK (π,.)πKπ,π,π‘,π β
β
πβπΊK (.,π)πKπ,π,π‘,π = πG
π,π,π‘,π + π PVπ,π,π‘,π
+ (πChπ,π,π‘,π β πDis
π,π,π‘,π ) β π¦π,π‘,π πππ,π,π‘,βπ, π, π‘, π (24)
β
πβπΊK (π,.)πK
π,π,π‘,π ββ
πβπΊK (.,π)πK
π,π,π‘,π = πGπ,π,π‘,π +πPV
π,π,π‘,π
+ πESSπ,π,π‘,π β π¦π,π‘,π π
ππ,π,π‘,βπ, π, π‘, π (25)
β π’π,π‘,π πK,maxπ β€ πK
π,π,π‘,π β€ π’π,π‘,π πK,maxπ ,βπ β πΊK , π, π‘, π (26)
β π’π,π‘,π πK,maxπ β€ πK
π,π,π‘,π β€ π’π,π‘,π πK,maxπ ,βπ β πΊK , π, π‘, π (27)
β€ πGπ,π,π‘,π β€ πG,max
π ,βπ β πΊEG, π, π‘, π (28)
β€ πGπ,π,π‘,π β€ πG,max
π ,βπ β πΊEG, π, π‘, π (29)
β€ πGπ,π,π‘,π β€ πMEG
π πG,maxπ ,βπ β πΊCN, π, π‘, π (30)
0 β€ πGπ,π,π‘,π β€ πMEG
π πG,maxπ ,βπ β πΊCN, π, π‘, π (31)
ππ,π,π‘,π β ππ,π,π‘,π β₯ 2(οΏ½ΜοΏ½πππKππ,π,π‘,π + οΏ½ΜοΏ½πππ
Kππ,π,π‘,π )
+ (π’π,π‘,π + πππ,π β 2)π,βπ, ππ β πΊK , π, π‘, π (32)
ππ,π,π‘,π β ππ,π,π‘,π β€ 2(οΏ½ΜοΏ½πππKππ,π,π‘,π + οΏ½ΜοΏ½πππ
Kππ,π,π‘,π )
+ (2 β π’π,π‘,π β πππ,π)π,βπ, ππ β πΊK , π, π‘, π (33)
min max
ππ,π‘,π ππ β€ ππ,π,π‘,π β€ ππ,π‘,π ππ ,βπ, π, π‘, π (34) 94Q. Zhang et al.
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2
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Fig. 2. A single line diagram of an example network with one damaged line.
Fig. 3. A virtual network created for the example network in Fig. 2.
Fig. 4. A crew assignment example with 2 depots and 5 crews.
2728
29
30313233
34
35
36
37
38
39
40
41
42
43
β
πββπΊB(l)
π’π,π‘,π β€ |πΊB(l)| β 1,βπ β πΊloop, π‘, π (35)
Constraints (24) and (25) are nodal power balance constraints ofactive and reactive powers, where πK
ππ,π,π‘,π and πKππ,π,π‘,π are active and
reactive power flows, and πGπ,π,π‘,π and πG
π,π,π‘,π are the power outputs of thegenerators. The active charging/discharging and reactive power out-puts of energy storage systems are denoted by πCh
π,π,π‘,π , πDisπ,π,π‘,π and πESS
π,π,π‘,π .Constraints (26)β(27) represent the active and reactive power limits ofthe lines, where the limits (πK,max
π and πK,maxπ ) are multiplied by the
line status binary variable π’π,π‘,π . Therefore, if a line is disconnected ordamaged, power cannot flow through it. Constraints (28)β(29) limit theoutput of the generators to πG,max
π and πG,maxπ . Similarly, the output of
the MEGs is limited in (30)β(31) if an MEG is installed (πMEGπ = 1).
Constraints (32) and (33) calculate the voltage difference alongline π between bus π and bus π, where ππ,π,π‘,π is the square of voltagemagnitude of bus π. The big-M method is used to relax constraints (32)and (33), if lines are damaged or disconnected. οΏ½ΜοΏ½ππ and οΏ½ΜοΏ½ππ are theunbalanced three-phase resistance matrix and reactance matrix of lineππ, which can be referred to [48]. The vector πππ,π represents the phasesof line ππ. Constraint (34) guarantees that the voltage is limited withina specified region (πmin
π and πmaxπ ), and is set to 0 if the bus is in an
outage area. Constraint (35) can guarantee the radiality network duringthe network reconfiguration. This model assumes that all the possibleloops can be identified by the depth-first search method. The set ofloops are given by πΊloop, and the set of switches in loop π is given by
πΊB(l). For each fuel-based generator, the total fuel consumption πΉπ,πis limited by the available fuel resources πFuelπ in constraint (36), asfollows:
πΉπ,π = πfβ
βπ‘
β
βππGπ,π,π‘,π β€ πFuelπ ,βπ β πΊG, π, π‘, π (36)
The operation constraints for ESSs and MESs include the change instate of charge (SOC), charging and discharging limits, and reactivepower limits. Let πΊES be the set of buses with ESSs, and πΊESC =πΊES βͺπΊCN.
πΈSOCπ,π‘,π =πΈSOC
π,π‘β1,π +
π₯π‘(β
βπ πChπ,π,π‘,π πCh β
β
βπ πDisπ,π,π‘,π βπDis)
πΈCapπ
,βπ β πΊESC, π, π‘, π (37)
πΈSOC,minπ β€ πΈSOC
π,π‘,π β€ πΈSOC,maxπ ,βπ β πΊESC, π‘, π (38)
0 β€ πChπ,π,π‘,π β€ βπ,π‘,π π
Ch,maxπ ,βπ β πΊESC, π, π‘, π (39)
0 β€ πDisπ,π,π‘,π β€ (1 β βπ,π‘,π )π
Dis,maxπ ,βπ β πΊESC, π, π‘, π (40)
βπESS,maxπ β€ πESS
π,π,π‘,π β€ πESS,maxπ ,βπ β πΊES, π, π‘, π (41)
Ch MES Ch,max
0 β€ ππ,π,π‘,π β€ ππ ππ ,βπ β πΊCN, π, π‘, π (42) 44Q. Zhang et al.
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27
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30
31
32
33343536373839404142434445464748495051525354
1
55
56s 57f 58
59
60
61s 62f 63s 64( 65M 66π 67a 68s 69
4 70
71s 72r 73E 74i 751 76πΆ 77T 78
0 β€ πDisπ,π,π‘,π β€ πMES
π πDis,maxπ ,βπ β πΊCN, π, π‘, π (43)
β πMESπ πESS,max
π β€ πESSπ,π,π‘,π β€ πMES
π πESS,maxπ ,βπ β πΊCN, π, π‘, π (44)
Constraint (37) calculates the state of charge (SOC) of ESSs (πΈSOCπ,π‘,π ).
Capπ is the maximum capacity of the storage system. To ensure safeSS operation, the SOC and charging (πCh
π,π,π‘,π ) and discharging (πDisπ,π,π‘,π )
ower of ESSs are constrained as shown in (38)β(40). Here, πΈSOC,minπ ,
SOC,maxπ , πCh,max
π and πDis,maxπ define the permissible range of SOC, and
aximum charging and discharging power, respectively. In constraints39)β(40), the binary variable βπ,π‘,π indicates that ESSs cannot chargend discharge at the same time instant. The ESS charging and discharg-ng efficiency are represented by πCh and πDis, respectively. The reactiveower of ESS, πESS
π,π,π‘,π , is kept within maximum limit, πESS,maxπ , through
onstraint (41). For MES units, the constraints (42)β(43) are presentedo that if πMES
π = 0, the output power is 0 at bus π. The same method ispplied for the reactive power in (44).
. Solution algorithm
When the number of scenarios is finite, a two-stage stochasticproblem can be modeled as a single-stage large linear programmingmodel, where each constraint in the problem is duplicated for eachrealization of the random data. As discussed before, the Monte Carlosampling technique can be used to generate a manageable number ofscenarios for problems where the number of realizations is too large orinfinite. In this work, the scenario decomposing method PH is used tosolve the proposed two-stage stochastic pre-event preparation problem.
3.1. Two-stage progressive hedging algorithm
The proposed two-stage stochastic pre-event preparation model (4)β(44) can be compactly reformulated as follows:
π = minπ₯,π¦π
ππ π₯ +β
βπ ππ(π )πππ π¦π (45)
s.t. (π₯, π¦π ) β ππ ,βπ (46)
In objective (45), the vectors π and ππ include the coefficientsrelated with the compact first stage variable π₯ and compact secondstage variable π¦π , respectively. The compact constraint (46) can en-sure the feasibility for solutions from each subproblem and scenario.When the non-anticipativity of the first stage variables is relaxed, thenthe PH algorithm decomposes the extensive form (EF) (45)β(46) intoscenario-based subproblems. Therefore, the proposed stochastic pre-event preparation problem with the total number π of scenarios canbe decomposed into π subproblems. In Algorithm 1, the proposedstochastic pre-event preparation problem is solved by PH algorithm.In Step 1, we initialize the problem. In Step 2β3, the subproblems withindividual scenarios are solved. In Step 4, we obtain the expected valueοΏ½ΜοΏ½ of the first stage solution by aggregating the solutions from Steps2β3. Step 5 calculates the value of the multiplier ππ . In Step 8, thesubproblems are solved by augmenting two terms: one linear term,which is proportional to the multiplier ππβ1π ; one squared two normterm of the difference between π₯ and οΏ½ΜοΏ½πβ1, which is penalized by πβπ.Steps 9β10 are similar as Steps 4β5. The algorithm terminates once allfirst stage decisions π₯π converge to a common οΏ½ΜοΏ½. Note that the two-stage model has been reformulated to a single-level problem for eachindividual scenario. In Algorithm 1, π is the iteration number, π is apenalty factor and π is the threshold value for termination.
pAlgorithm 1 PH Algorithm for Solving Stochastic Pre-event PreparationProblem1: Initialization: the iteration π.2: For each individual scenario π β π, solve.3: π₯(π)π βΆ= argminπ₯{ππ π₯ + πππ π¦π βΆ (π₯, π¦π ) β ππ }.4: οΏ½ΜοΏ½(π) βΆ=
β
βπ βπ ππ(π )π₯(π)π .5: π(π)π βΆ= π(π₯(π)π β οΏ½ΜοΏ½(π)).6: π βΆ= π + 1.7: For each individual scenario π β π, solve.8: π₯(π)π βΆ= argminπ₯{ππ π₯+ πππ π¦π + π(πβ1)π π₯+ π
2βπ₯(π)π β οΏ½ΜοΏ½(π)β2 βΆ (π₯, π¦π ) β ππ }.
9: οΏ½ΜοΏ½(π) βΆ=β
βπ βπ ππ(π)π₯(π)π .10: π(π)π βΆ= π(πβ1)π + π(π₯(π)π β οΏ½ΜοΏ½(π)).11: if ββπ βπ ππ(π )βπ₯(π)π β οΏ½ΜοΏ½(π)β β€ π then12: Go to Step 5.13: else14: terminate.15: end if
Algorithm 2 Multiple Replication Procedure1: Initialization: Set πΌ β (0, 1) (e.g., πΌ = 0.05), sample size π,
replication size ππ and a candidate solution οΏ½ΜοΏ½ β π.2: For π = 1, 2, ..., ππ .3: Sample i.i.d. observations ππ1 , ππ2 , ..., πππ from the distribution of π .4: Solve (SPn) using ππ1 , ππ2 , ..., πππ to obtain π₯πβπ .5: Calculate πΊπ
π (οΏ½ΜοΏ½) βΆ= πβ1βπ
π=1(π (οΏ½ΜοΏ½, πππ ) β π (π₯πβπ , πππ )).
6: End for.7: Calculate gap estimate πΊπ(ππ) βΆ=
1ππ
βπππ=1 πΊ
ππ (οΏ½ΜοΏ½).
8: Calculate sample variance π 2πΊ(ππ) βΆ=1
ππβ1βππ
π=1(πΊππ (οΏ½ΜοΏ½) β πΊπ(ππ))2.
9: Let π βΆ= π‘ππβ1,πΌππΊ(ππ)ββ
ππ .10: Obtain one-sided CI on [0, πΊπ(ππ) + ππ].1: Output: Approximate (1βπΌ) as the level confidence interval on ποΏ½ΜοΏ½.
3.2. Convergence analysis and solution validation
As shown in Algorithm 1, the convergence metric ππ for the progres-ive hedging algorithm at each iteration π is expressed as the deviationrom the mean summed across all first stage variables π₯π (π) and the
average value of the first stage variable οΏ½ΜοΏ½π as follows:
ππ =β
π βπππ(π )βπ₯π (π) β οΏ½ΜοΏ½πβ (47)
Since the solution is obtained using a limited number of damagecenarios, the quality of the solution requires verification. Adoptedrom [49], the MRP can be applied to repeat generating π scenarios andolving the proposed model for π times. Then the confidence intervalCI) is constructed to calculate the optimality gap. The detailed steps inRP are shown in Algorithm 2, where πΊπ(ππ) is the gap estimate and
2πΊ(ππ) is the sample variance. Numerical results for the convergencenalysis and solution validation of the test case are given in the nextection.
. Case study
This section uses a large-scale system as a test case to verify thecalability and effectiveness of the two-stage stochastic pre-event prepa-ation model. This large-scale system consists of 3 existing test systems,PRI ckt5, ckt7 systems [50], and IEEE 8500 bus system [51], Follow-ng the suggestions from [15], the unit costs in the simulation are πΆD =4$/kWh for load shedding at all buses, πΆSW = 8$ for each line switch,F = 1$/L and πF = 0.3 L/kWh for fuel consumption of generators.he Pyomo and Gurobi mixed-integer solver [52] are used to solve theroposed stochastic model. All experiments are implemented on the
79Q. Zhang et al.
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3
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Fig. 5. Resource allocation of large-system with the proposed model.
Fig. 6. The convergence metric comparison with and without soft-start solutions.
91011121314151617
Iowa State University Condo cluster, whose individual blade consists oftwo 2.6 GHz 8-Core Intel E5-2640 v3 processors and 128 GB of RAM.
4.1. Pre-event preparation results
This case study include 9 depots that are hosting a total of 27 crews,9 dispatchable DGs, 8 MEGs, 3 MESs, 123 switches, 5 small PVs, 15large PVs, and 12 ESSs. The active and reactive power capacities of the9 DGs are 300 kW and 250 kVAr. The active power capacity of smallPVs ranges from 11 kW to 22 kW. The active power capacity of large
PVs is 500 kW. The 12 ESSs are rated at 500 kW/ 3500 kWh. The pre-event preparation model of the large-scale system is solved in 10.2 hwith 10 damage scenarios. The locations of MEGs, MESs, and numberof crews are shown in Fig. 5. 27 crews are allocated to 9 differentdepots. The value inside the crew depot in Fig. 5 represents the numberof crews assigned to that depot. Areas with a large number of crewsindicate that the lines in the area have high damage probabilities.
As discussed in Section 3.2, the convergence metric can be used toevaluate the convergence speed of the proposed model. At the sametime, the computational speed with and without a soft-start solution are
18Q. Zhang et al.
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Fig. 7. Aggregated damaged areas.
Fig. 8. Comparison between the base model and the proposed method.
15161718192021222324252627
compared. In this paper, a soft-start solution means that the previouslycomputed solution in other instances will be used as the starting point.The comparison result is shown in Fig. 6. If the convergence metricreaches the convergence threshold of 0.01, the algorithm will stopand obtain the optimal solution. The instance with a soft-start solutionconverges at 57 iterations and takes 10.2 h. The case without a soft-start solution converges after 100 iterations and takes 24.3 h. Totest the solution quality with MRP, based on the limited number ofgenerated damage scenarios, the one-sided CI of the obtained solutionis [0, 12.48%]. This small gap indicates that the damage scenarios arerepresentative and the solution is stable with high quality.
To evaluate the performance of the developed pre-event preparationmodel, the model is compared to a base model. The base case isgenerated by the following steps: (i) one MEG is pre-positioned at each
substation. (ii) Extra MEGs are pre-positioned at high-priority loads.(iii) PV and ESS are not considered. (iv) Fuel is allocated to the MEGssuch that the MEGs can operate for at least 24 h. (v) Crews are allocatedevenly between depots. In this work, the average outage duration iscalculated by dividing the sum of outage duration for the loads bythe total number of loads. To compare the proposed model and thebase model, a random scenario is generated and test the response ofthe system. The generated scenario has 103 damaged lines, which areaggregated to 34 damaged areas in Fig. 7. Each circle represents therepair time needed for the specific damaged area considering all theaggregated damaged lines.
The comparison between the base model and the proposed methodis shown in Fig. 8. In the base model, the total restored energy is231,422.38 kWh and the average outage duration is 14.69 h. In the
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Fig. 9. Pre-event resource allocation results with different PV penetration levels.
141516171819202122232425262728
proposed method, the total restored energy is 291,727.48 kWh and theaverage outage duration is 11.28 h. Therefore, approximately 20.67%more loads are served by the proposed method and the outage durationdecreased by 30.22%.
4.2. Impacts of solar PV on system resilience
To show the advantages of the PV systems, the responses of thesystem with the proposed pre-event preparation method and differentPV penetration levels are tested. Three rated capacities of PV systemsare considered: (i) Capacity 1 PV, which represents residential PVpanels and the rated capacity is assumed to be 6 kW; (ii) Capacity 2 PV,which represents mid-size PV systems and the rated capacity is assumedto be 48 kW; (iii) Capacity 3 PV, which represents large utility PV farm
and the rated capacity is assumed to be 2000 kW. Based on the numberof different types of PVs, 6 PV penetration levels are defined as 9%,27%, 45%, 63%, 81%, and 99%. The number of Capacity 1, 2, and 3PVs for each PV penetration level is summarized in Table 1. To bettercollaborate the setting of PV penetration, the number of dispatchableDGs has been changed to 10 and the positions of those DGs have beenchanged accordingly.
Based on the results of Fig. 9, it can be observed that differentPV penetration levels have different allocation results for the flexibleresources, including the positions of MEGs, MESs, and the number ofrepair crews.
Fig. 10 shows the percentage of power served during the event, andafter the repair process starts. Tables 2 and 3 compare the amount ofload served and average outage duration with different levels of PVpenetration.
Based on the results from Fig. 10, Table 2, and Table 3, it can
be seen that the penetration of PV contributes to enhancing system 29Q. Zhang et al.
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Fig. 10. Load served percentage comparison of the proposed model with various PV penetration levels and the base model.
6
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10111213141516171819202122232425262728293031323334353637383940414243
44
45
able 1V penetration levels and the number of PV systems with different rated capacities.PV penetrationlevel
Capacity 1PV
Capacity 2PV
Capacity 3PV
9% 8 1 127% 24 4 345% 40 7 563% 63 9 781% 72 12 999% 88 15 11
Table 2The amount of load served and resilience improvement with different PV penetrationlevels.
PV penetrationlevel
Load served(kWh)
Resilience improvementpercentage (%)
0 251,210.72 β9% 318,668.37 26.8527% 335,525.77 33.5645% 336,710.74 34.0463% 344,588.22 37.1781% 360,668.04 43.5799% 364,785.93 45.21
Table 3The amount of average outage duration and outage decreased percentage with differentlevels of PV penetration.
PV penetrationlevel
Average outageduration (h)
Outage decreasedpercentage (%)
0 14.69 β9% 12.33 16.0727% 11.72 20.2245% 11.65 20.6963% 11.21 23.6981% 10.45 28.8699% 10.12 31.11
resilience. Approximately 31.13% more loads are served than the basemodel when the proposed method with 99% PV penetration is used.Also, the average outage duration decreased by 31.12%. However,compared with 81% PV penetration level, the proposed method with99% PV penetration does not have significant improvement.
5. Conclusion
Extreme weather events may severely impact the electric grid infras-tructures, causing major damage and faults in the system. This leads topower outages for an extended period. It is up to the electric utilityto plan how to prepare for such an event and restore power to thecustomers after the event. When an extreme weather event hits thedistribution system, the damaged network may hinder the physicaldelivery of mobile resources and repair crews. In addition, withoutproper preparation, utilities will be overwhelmed with the numberof tasks that must be conducted, including assigning tasks to crews,managing crews coming from different areas, and dispatching portablegenerators to supply critical customers. Therefore, to achieve fast andefficient response, it is critical to pre-position crews, equipment, andother resources before the severe event occurs. In this paper, a two-stage stochastic pre-event preparation and resource allocation methodis proposed for upcoming extreme weather events, which enhancesthe system resilience and enables more efficient post-event restoration.The proposed pre-event method leverages the pre-allocation of mobileresources, fuel resources, and labor resources. By considering differentoperation modes of distributed PV systems, the proposed model alsofacilitates the benefits of solar powers in the resilience improvement ofdistribution grids. According to the case studies, the following obser-vations are found: (i) Compared to the base model without pre-eventresource allocation, the proposed pre-event preparation model canserve more loads and reduce the outage duration. (ii) Based on theresponse of the system with different PV penetration levels, it canbe observed that the proposed pre-event preparation model with highPV penetration can further improve system resilience and reduce theoutage duration. Therefore, PV systems can play a critical role inimproving distribution grid resilience and further promote renewableenergy deployment. (iii) By considering the trade-off between solutionaccuracy and computation efficiency, the result of MRP indicates thatthe proposed modelβs solutions with a limited number of scenarios canbe very stable and of high quality. The scalability of the proposedpre-event preparation model is verified with a large-scale system. Thetrade-off between the cost of pre-event resource allocation and therisk associated with damage loss will be considered under upcomingextreme weather events in future work.
CRediT authorship contribution statement
Qianzhi Zhang: Conceptualization, Methodology, Software, Writ-
ing - original draft, Validation. Zhaoyu Wang: Supervision, Project 46Q. Zhang et al.
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administration, Funding acquisition. Shanshan Ma: Writing - review& editing, Software, Validation, Methodology. Anmar Arif: Writing -review & editing, Software, Validation, Methodology.
Declaration of competing interest
The authors declare that they have no known competing finan-cial interests or personal relationships that could have appeared toinfluence the work reported in this paper.
Acknowledgment
This work was supported by the U.S. Department of Energy WindEnergy Technologies Office under Grant DE-EE0008956.
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