Risk-informed Resilience Planning of Transmission Systems Against Ice Storms

1 Risk-informed Resilience Planning of T ransmission Systems Against Ice Storms Chenxi Hu, Student Member , IEEE , Y ujia Li, Member , IEEE, , Y unhe Hou, Senior Member , IEEE, Abstract —Ice storms, known for their sev erity and predictability , necessitate proacti ve resilience enhancement in po wer systems. T raditional approaches often overlook the endogenous uncertainties inherent in human decisions and underutilize p redictiv e inf ormation like for ecast accuracy and preparation time. T o bridge these gaps, we proposed a two-stage risk-inf ormed decision-dependent resilience planning (RIDDRP) for transmission systems against ice storms. The model leverages predictive information to optimize resour ce allocation, considering decision-dependent line failure uncertainties introduced by planning decisions and exogenous ice storm-related uncertainties. W e adopt a dual-objective approach to balance economic efficiency and system resilience across both normal and emergent conditions. The first stage of the RDDIP model makes line hardening decisions, as well as the optimal sitting and sizing of energy storage. The second stage evaluates the risk- informed operation costs, considering both pr e-event preparation and emergency operations. Case studies demonstrate the model’ s ability to leverage predictive information, leading to more judicious in vestment decisions and optimized utilization of dispatchable resour ces. W e also quantified the impact of different properties of predicti ve inf ormation on resilience enhancement. The RIDDRP model provides grid operators and planners valuable insights for making risk-inf ormed infrastructure inv estments and operational strategy decisions, ther eby impro ving prepar edness and response to future extreme weather events. Index T erms —Resilience enhancement, decision-dependent un- certainty , transmission system planning, stochastic optimization. N O M E N C L AT U R E Sets and Indices L , ij Set and index of lines. N , i Set and index of buses. S , s Set and index of scenarios. T , t Set and index of time. P arameters ∆ t T ime interv al. η + /η − Charging / Dischar ging efficienc y of the energy stor- age. θ i,t,s Angle of bus i at time t in scenario s . B ij Susceptance of line ij . C B Energy storage in vestment budget. c ch/dis Charging / Discharging cost of the energy storage. C L Line in vestment budget. c N/E LS Cost of load shedding under normal / extreme condi- tion. c b Unit in vestment cost of the ener gy storage. c g Operation cost of generator g . c l In vestment cost of hardening line l c w W ind curtailment cost of wind farm w . P Max / Min g Maximum / Minimum output of generator g . P Max ij Maximum capacity of line ij . This work was supported by National Key R&D Program of China (2023YF A1011301), Joint Research Fund in Smart Grid (Grant No. U1966601) under cooperati ve agreement between the National Natural Science Foundation of China (NSFC) and State Grid Corporation of China (SGCC), National Natural Science Foundation of China (NSFC) under Grant 52177118. P D,N /E i,t,s Load on bus i at time t under normal / extreme condition in scenario s . p s Probabilities of each scenario s . V ariables α i /β i Load shedding ratio of bus i under normal and emergenc y scenarios. γ w,t,s Binary variables indicating the status of wind farm w at time t in scenario s . (1 if wind farm w is damaged; 0 otherwise). µ ij,t,s Binary variables indicating the status of line ij at time t in scenario s . (1 if line ij is damaged; 0 otherwise). ε w ij,t,s /o Binary variables indicating the status of line ij at time t in scenario s with / without line hardening. (1 if line ij is damaged; 0 otherwise) P ch,N/E i,t,s Charging po wer of the energy storage at bus i at time t under normal / extreme condition in scenario s . P dis,N/E i,t,s Discharging po wer of the energy storage at bus i at time t under normal/e xtreme condition in scenario s . P g ,N/E i,t,s Activ e power output of the generator at bus i at time t under normal / extreme condition in scenario s . P LS,N/E i,t,s Operation cost of generator g . P w,N /E i,t,s Forecasted wind outputs on bus i at time t under normal / extreme condition in scenario s . P wc,N /E i,t,s W ind curtailment on bus i at time t under normal / extreme condition in scenario s . P N/E ij,t,s Activ e power on line ij at time t under normal / extreme condition in scenario s . S N/E i,t,s State of char ge of the energy storage on bus i at time t under normal / extreme condition in scenario s . x ch,N/E i,t,s Binary variables indicating the dischar ging status of the energy storage on b us i at time t under normal / e xtreme condition in scenario s . (1 if the energy storage is charging; 0 otherwise). x dis,N/E i,t,s Binary variables indicating the dischar ging status of the energy storage on b us i at time t under normal / e xtreme condition in scenario s . (1 if the energy storage is discharging; 0 otherwise). x ij Binary variables indicating the hardening decision of line l . (1 if line ij is hardened; 0 otherwise). Z i Capacity of the ener gy storage at b us i . I . I N T R O D U C T I O N Power systems are vulnerable to weather-dri ven catastrophes, such as hurricanes, ice storms, and floods, resulting in significant economic loss and heightened risks to public safety [1]. Among them, ice storms are note worthy due to their long duration and wide scope. Ice accretion and heavy snow can cause system failure from line overloading, galloping, and icing flasho ver . Ad- ditionally , the strong winds combined with ice accumulation can greatly influence power generation and cause major disruptions, especially with the increasing penetration of renew able ener gy 2 sources [2]. In 2008, the ice disaster in Southern China led to more than $50 billion in economic losses and the failure of thousands of power lines and to wers [3]. More recently , T exas was struck by the winter storm Uri in February 2021, resulting in $130 billion in losses and about 2000 MW wind power went offline [4]. There is a gro wing concern that the frequency and sev erity of such e vents could increase as a result of climate change, increasing the exposure of po wer systems to failures [5]. This emphasizes the importance of enhancing po wer system resilience on the ice disaster prev ention capabilities. During ice storms, transmission lines are susceptible to ice accretion which can lead to large-scale blackouts. Effec tiv e resilience enhancement measures include the application of anti- icing materials, the use of de-icing technologies, and the de velop- ment of de-icing operation strate gies [6]. Apart from operational strategies, this research emphasizes the importance of incorpo- rating resilience considerations as early as the planning stage. T ypical strategies include robust structure design, or relocating facilities to make the system less prone to disruptive ev ents. A key hardening strategy against ice storms is the use of anti- icing coating materials [7], which enables the transmission line to resist thicker ice. Additionally , strategically relocating facilities, such as bulk energy storage, can not only optimize the system operation under normal conditions b ut also help pro vide a con- tinuous electricity supply for critical loads during outages. The distributed characteristics of ener gy storage as well as its load- shifting capability make it naturally suitable for normal operation and resilience enhancement during emer gencies [8]. While those measures can reduce the chance of component failures and the needed restoration efforts, hardening and upgrading the entire system is expensiv e. Therefore, how to prioritize transmission lines for hardening, and how to allocate the ener gy storage to balance costs and system resilience hav e become great challenges that require further research. Despite numerous resilience enhancement measures, achie ving effecti ve resilience planning must be crafted with a wareness and understanding of the characteristics of different kinds of system failure. Those dif ferences mainly lie in two aspects [9]: (1) how much predicti ve information, e.g., e vent types, start time, duration and trajectory , system operators have to make prepa- rations; (2) ho w much of the systems remain controllable and operable during contingencies. While resilience enhancement with limited resources has been widely in vestigated [10], [11], existing research seldom underlines the effect and potential of predictiv e information on the resilience enhancement of po wer systems. W ith advanced forecasting technologies, we are able to predict extreme weather e vents, including ice storms [12]. This predicti ve information is pi votal for preparatory , pre venta- tiv e, and remedial actions. In this context, the term predictive information is defined as the information that quantifies insights deriv ed from available information to anticipate future ev ents [13]. The lead time and accuracy of predictive information can influence the operational decision-making process by furnishing valuable insights before ev ents, thereby f acilitating preparation and engendering v aried response strategies during contingencies. Howe ver , much of the existing research often treated predictiv e information as deterministic inputs, overlooking the inherent uncertainty of the information itself. Ma et al [14] proposed a tri-lev el resilience enhancement strategy by incorporating the coupling of the hardening decisions and the damage uncertainty . Li et al [15] proposed a scenario-wise decision-dependent ambi- guity set to model the uncertainty brought by preventi ve resilient enhancement strategies and the related e xtreme weather-induced system contingencies. In fact, given the current technological capabilities and anticipated advancements, the accuracy and re- liability of information can v ary significantly . The availability of information can greatly influence the decision-making process, leading to diverse in vestment strategies. In this re gard, a compre- hensiv e risk-informed planning model should not only account for uncertainties stemming from in vestment decisions but also consider the uncertainties intrinsic to predictiv e information. Beyond the previously mentioned hardening measures, the impact of the planning decisions on the operating states of the power system is another usually overlook ed yet significant part. Previous resilience planning research mainly focused on emergenc y scenarios during extreme ev ents, ignoring the pre- ceding risk-informed conditions wherein the e vents are already anticipated, given certain predictive information. Though some research has encompassed operation during various operational conditions, a primary concern is that these methods neglect the transition in operational objectives between normal and emer- gency conditions. Hossein et al. [16] proposed a resilience plan- ning model for transmission systems considering both normal and emergenc y conditions using distributed energy resources. Lagos et al. [17] proposed a framew ork for resilient network in vestments against earthquak es using an optimization via sim- ulation approach. Though their models acknowledge v arying operational states, these conditions are treated independently and the coordination between the normal and emer gency states is not considered. During contingencies, there is a paradigmatic shift in operational objectives. The priority transitions from economic operation to an emphasis on retaining critical loads, rather than ensuring the adequacy of all loads. This significant change underscores the inadequacy of traditional planning approaches and highlights the necessity for a coordinated strategy that can pre-dispatch av ailable resources. Based on the previously discussed research gaps, a two-stage risk-informed decision-dependent resilience planning (RIDDRP) model for transmission systems against ice storms is proposed in this study . The main contributions are summarized as follo ws: • Our model quantifies the impact of predicti ve information on resilience planning, underscoring its criticality . Specif- ically , two distinct aspects are incorporated into the RID- DRP model: the time-varying accuracy and the anticipatory preparation time prior to ice storms. This integration allo ws for more optimized utilization of av ailable resources to enhance system resilience. • Considering the changes in operation objectives under nor- mal conditions in anticipating ice storms and emer gency conditions during ice storms, our model emphasizes risk- informed coordination between those conditions to achieve judicious in vestment decisions. A dual-objecti ve approach is employed to achiev e trade-offs between economy and resiliency . Thus, a voiding the o ver -design of the system and saving unnecessary costs. • The inherent uncertainty of predictive information, i.e., the time-varying accuracy and the associated timing of antici- patory preparations, the DDU introduced by line hardening decisions, and various ice storm-related e xogenous uncer - tainties are incorporated and formulated. By addressing these multifaceted uncertainties, our model can characterize their coupled effect on the system operation under ice storms to avoid potential insecure issues. • A two-stage risk-informed decision-dependent resilience planning (RIDDRP) model with mixed-inte ger recourse is constructed. The first stage makes line hardening decisions, as well as the optimal sitting and sizing of ener gy storage. The second stage ev aluates the risk-informed operation 3 costs. T o relie ve computational burdens, the progressi ve hedging algorithm (PHA) is adopted to solv e the lar ge-scale mixed-inte ger linear programming problem. T o v alidate the efficac y of the proposed model, comparative analyses are conducted under various anticipatory preparation times. It is shown that the proposed resilience enhancement strategy can bolster system resilience enhancement by adopting more judicious in vestment and optimizing the utilization of dispatchable resources. The remainder of this paper is organized as follows: Section II introduces the system modelling under ice storms. Section III quantifies the impact of predictive information. Section Iv describes the mathematical formulation of the proposed two- stage RIDDRP model and the solution algorithm. Section IV discusses the solving algorithms as well as the lower bound guarantee. Section V presents case studies and Section VI concludes this study . I I . T H E F R AG I L I T Y M O D E L O F T R A N S M I S S I O N S Y S T E M U N D E R I C E S T O R M S A. Ice storms-induced System Outag e In addressing power transmission system vulnerabilities under various weather intensities, we adopt the concept of the fragility curve to model the relationship between the failure probability of system components and the weather intensity . Under ice storms, the weather intensity z is denoted by the ice thickness of ice r . Gi ven the meteorological data, a freezing rain ice accretion model is utilized to calculate ice loads [18]: r = h ρ i π p ( P r ρ w ) 2 + (3 . 6 v L ) 2 (1) where r represents the ice thickness, P r and L denote the precipitation rate and the liquid water content, respectiv ely . The follo wing empirical formulation is utilized to model their relationship: L = 0 . 067 P r 0 . 846 (2) where h denotes the hours of freezing. ρ i and ρ w are the density of ice and water with 0 . 9 g /cm 3 and 1 g /cm 3 , respectiv ely . v is the wind speed. Historical statistic analysis can be conducted to estimate proper probability distribution functions (PDFs) for each weather-related parameter . In this study , PDFs stated in [19] are adopted to model ice storms. As an ice storm propagates through the transmission system, various transmission corridors will be affected. T o simplify the problem, we assume that all lines share the same fragility curve. The failure probabilities are independent of each other for a fixed weather intensity . Then, the f ailure probability of a component can be expressed as: P f ( r ) =        0 r < R exp  0 . 6931( r − R ) 4 R  − 1 R ≤ r < 5 R 1 r ≥ 5 R (3) where R is determined by the line hardening decision x ij . Considering the spatial and temporal changes of ice storms, a transmission corridor is divided into L line segments and a failure in a single segment will result in the failure of the entire corridor due to their connection in series. The total duration of the ice storm T , is divided into N time steps with a shorter duration period ∆ t , Then, the failure probability of the k -th segment in the i -th corridor at time t j can be ev aluated once giv en the meteorological data. Due to the series connection, the cumulative failure probability of the transmission corridor connecting bus i and j during the ice storm period T can be ev aluated using all its se gments: P f ij ( r ) = 1 − L Y l =0 (1 − P f l ( r )) (4) where P f l ( r ) is the failure probability of the l -th segment and P i ( r ) is the cumulative failure probability of the i -th transmis- sion corridor . B. Decision-dependent Line F ailur e Pr obability In this study , the endogenous uncertainty lies in the realization of contingencies influenced by the in vestment decisions made in the planning stage. Specifically , the ability of a transmission line to resist ice accretion can be manipulated through purposeful hardening designs. Consequently , the uncertain line damage status µ ij,t under a certain ice storm scenario is interrelated with the line hardening decisions and becomes DDU. In this model, the empirical fragility function is adopted to estimate the failure probability of a giv en transmission line P f , which is parameterized by both the endogenous hardening decisions x and the exogenous intensity of ice storms. Under the same ice thickness, the failure probability of a transmission line will decrease if it is hardened. Since the potential number of x will gro w exponentially with the size of the system, it would be unrealistic to generate scenario sets that contain all possible realizations of x . T o tackle this problem, we decompose the outcome of a random ev ent µ ij,t,s , i.e., whether the line is damaged by ice storms, into two conditional independent binary variables ϕ w ij,t,s and ϕ o ij,t,s , which represent the line damage status with or without line hardening respectively . The damage status variables will be set as 0 until line damage occurs. µ ij,t,s = (1 − x ij,s ) ϕ o ij,t,s + x ij,s ϕ w ij,t,s (5) Thus, the line status under a certain scenario µ ij,t,s can be represented as an explicit function of the first-stage line hardening decisions x . Then, ϕ w ij and ϕ o ij can be determined in adv ance during the scenario generation process to handle the uncertainty of line damage status. Therefore, we can proacti vely mitigate line damage risk through strategic hardening decisions. T o model the inter-temporal correlation during the repair process, the line status is treated as a Bernoulli process and we take the first time stamp t 0 / 1 that ϕ o/w ij,t,s = 1 as the initiation of line damage, which can be sampled from the fragility curve 3. During the repairing process that lasts T r hours, the line status will be fixed as 1 to represent line outages. If the line is repaired before the end of the ice storm, we assume that this line will not be damaged again in the remaining time of the ice storm. The whole process of determining the line status µ ij,t,s can be illustrated in Fig. 1. By incorporating DDU during planning, we are able to in- corporate not only the exogenous intensity of ice storms but also the endogenous line strength to withstand certain levels of ice accretion, thereby further releasing the system’ s potential for resilience enhancement in the face of extreme e vents. C. Ice Storm-r elated Exogenous Uncertainties 1) Uncertainty of wind turbine blade fr eezing: During ice storms, ice accumulation can lead to the failure of wind turbines. In our model, the f ailure probability of a wind turbine follo ws log-logistic distribution for a given ice thickness r : P f W T ( r ) = ( r /α ) β 1 + ( r /α ) β (6) 4 Ha r d e n i n g E f f e c t Fail ure Probab i l i t y Wi t hout hardeni ng 𝒙 𝒊𝒋 , 𝒔 = 𝟎 Wi t h hardeni ng 𝒙 𝒊𝒋 , 𝒔 = 𝟏 1 0 𝝁 𝒊𝒋 , 𝒕 , 𝒔 = 𝝓 𝒊𝒋 , 𝒕 , 𝒔 𝒐 𝑻 𝒓 𝒕 𝟎 𝒕 1 0 𝝁 𝒊𝒋 , 𝒕 , 𝒔 = 𝝓 𝒊𝒋 , 𝒕 , 𝒔 𝟏 𝑻 𝒓 𝒕 𝟏 𝒕 Ic e T hi ck ness Fig. 1. Decision-dependent damage status of line ij with/without hardening decision. where α and β are both parameters of the log-logistic distribution that can be estimated using historical data [20]. 2) Uncertainty of Load: Given the historical load profile L H , we assume a homogeneous load pattern prior to and during the ice storm. The load uncertainty le vel κ i,s that follows a normal distribution is utilized to generate load profiles for a stochastic scenario [14]: Load D i,t,s = κ i,s L H (7) 3) Uncertainty of r epair time: The repair time of each dam- aged component T r ( s ) , is a random variable that depends on dispatches of repair crews and resources. In this study , we assume that the repair time is independent and follows the same W eibull distribution [21]. The damaged status of the line or wind turbine will last until the end of the repair time. For each damaged line ij or wind turbine w , the repair time is: T r ij /w ( t ) =      β α  t α  β − 1 e − t β /α β t ≥ 0 0 otherwise (8) where α = 4 and β = 10 are parameters of the distribution. I I I . Q U A N T I FI C A T I O N O F T H E I M PAC T O F P R E D I C T I V E I N F O R M A T I O N In this section, we delve into the critical role of predictiv e information. Specifically , our focus is on quantifying the impact of predictiv e information through two key concepts: the time- varying accuracy and the anticipatory preparation time. The former aims to model the ev olving accuracy , and the latter aims to model its operational impact ov er time. By mathematically modeling the impact of predictiv e information, we aim to inte- grate those characteristics into the decision-making process for resilience enhancement. A. T ime-varying accuracy quantification Giv en the ability to forecast ice storms, operators can obtain critical predicti ve information prior to these e vents, enabling proactiv e preparation. Howe ver , the inherent uncertainty in the predictiv e information, such as the forecasting accuracy , has seldom been considered and properly modeled. By incorporating time-sensitiv e information, the uncertainty and a vailability of resources can be considered in the model, which can significantly influence the dispatch of hybrid resources for better preparation and response against ice storms. This, in turn, can lead to the development of more ef fective operation strategies adapted to the e volving circumstances. As the e vent approaches, the uncertainty f acilitating proacti ve preparation tends to decrease, enabling more precise preparatory measures. This phenomenon 𝑡 1 𝑡 2 𝑡 3 T im e Dimensions of U nc e r ta int y Unc e rta int y R e g ion T ra jec tor y of pre dictive e ve nts Fig. 2. Illustrative e xample of the uncertainty level e volution in predictiv e ev ents. can be mathematically encapsulated by a monotonically decreas- ing characteristic. U ( X t +1 ) ≤ U ( X t ) , ∀ t (9) where X t denotes the predictiv e information at time t and U represents the uncertainty le vel of the current information. The associated confidence intervals can be expressed as: C I ( X t ) = [ µ t − z σ t , µ t + z σ t ] (10) where µ t and σ t are the mean and standard deviation of the predictiv e information at time t , and z is the z-score correspond- ing to the desired confidence le vel. The decreasing tendency of the uncertainty associated with the predictiv e information can be characterized by the gradually narrowed confidence intervals C I ( X t ) as time approaches the occurrence of the anticipated ev ents. An illustrati ve example of the e volutionary trajectory of uncertainty lev els in predictive events is sho wn in Fig.2. Although numerous posterior indices exist to quantify the accuracy of predicti ve information, preempti vely estimating this accuracy remains a challenge. In this study , we navigate this complexity by mapping the uncertainty to a time-v arying pre- venti ve load-shedding penalty , denoted as c N LS , before the on- set of ice storms. This can be achiev ed through a generally monotonically decreasing function, which serves to quantify the progressiv ely diminishing uncertainty of predicti ve information as the ev ent approaches. This approach is grounded in the ratio- nale that initiating proactiv e measures at an earlier stage can be riskier since there is higher uncertainty for future conditions. The decisions made under current conditions may not remain v alid or adv antageous in the future, thereby elev ating the associated risks and potential costs of load shedding. It is note worthy that other types of functions can also be selected based on the specific problem at hand. B. Anticipatory pr eparation time As mentioned above, predictiv e information enables pre- venti ve strategies against ice storms. Ho wever , differences in operational strategies under normal and emergency conditions should be ackno wledged. T raditional planning objecti ves aim at ensuring resource adequacy , which can lead to resource depletion during emergencies. In contrast, the objecti ve of resilience plan- ning requires the consideration of both normal and emergency states. Under normal conditions, the objectiv es align with tradi- tional planning, focusing on resource adequac y . Ho wever , during emergencies, the objective shifts to maximizing the critical load that can be retained by coordinating resources across different stages. The change of operational objectiv es has seldom been considered in past research. T o present a more holistic and rob ust planning strategy , in this study , both the normal operation conditions prior to ice storms and the emergenc y conditions during extreme events are 5 incorporated as the planning objecti ve to strike a balance between economic efficiency and system resilienc y . A critical aspect of this strategy is the concept of anticipatory pr eparation time , which refers to the period before an ice storm when preparations are initiated based on predictiv e information. Given the av ail- able predicti ve information and its associated uncertainty lev el, operators can select an appropriate time to start preparations for anticipated ice storms. This decision is crucial as it allo ws for the strate gic pre-dispatch of resources, including charging energy storage, to mitigate the risk of large-scale load shedding during ice storms. The determination of this commencing time is influenced not only by the need for operational strategy adjustments but also by the capabilities of current forecasting technology . Due to the time-varying uncertainty associated with pre- dictiv e information, which we have mapped to a pre venti ve load-shedding penalty that decreases with time, the anticipatory preparation time is directly linked to the economic risks. The earlier the preparation anticipatory preparation time, the higher the economic risks. Therefore, operators must judiciously choose the start time for these preparations, balancing the benefits of early action against the escalating economic risks posed by prolonged periods of uncertainty . In this study , the anticipatory preparation time is parameterized by ξ , representing the time interval between the initiation of risk-informed operation and the occurrence of the ice storms. V arious values of ξ will be selected in the case study to compare and explore the impact of this parameter on system planning and operations. Considering the transition from normal conditions to e xtreme ev ents, the set of normal operating conditions and the emergenc y operating conditions is determined by the advanced time determined by the predicted information. Since the duration of extreme ev ents is an exogenous factor affected by nature, T E can be viewed as a fixed set. The set of risk-informed normal operation time T N duration is parameterized by the advanced time ξ influenced by the technology lev el and the operator’ s decisions [12]. Thus, there is: T N = { t t | i = t E − ξ , . . . , t E start − 2 , t E start − 1 } (11) T E = { t t | i = t E start , . . . , t E end } (12) Subsequently , we adopted a dual objectiv e to model the changes in the operational strategies: ψ s ( T ) = ψ N s ( T N ) + ψ E s ( T E ) (13) where ψ s is the total cost given a specific scenario s , ψ N s and ψ E s are the associated normal and emergent operational cost, respectiv ely . Given the circumscribed anticipatory preparation time, this approach empowers operators to dispatch av ailable resources more efficiently , thereby furnishing insights during the planning stage. I V . M A T H E M A T I C A L F O R M U L A T I O N O F T H E T W O - S TAG E R I S K - I N F O R M E D R E S I L I E N C E P L A N N I N G M O D E L In this section, we propose a tw o-stage stochastic mixed- integer programming model for risk-informed resilience planning considering ice storms, as shown in Fig.3. The first stage is to make system planning decisions, i.e., line hardening as well as sitting and sizing of energy storage. The second stage is for operational cost e valuation under ice storm scenarios, which contains normal conditions prior to and emer gent conditions during ice storms. By jointly optimizing the entire operational cost before and during the e vents, we aim to achieve the trade- E m er g en t Co n d i t i o n Objecti v e : Maxi m i ze ret ai n ed cri t i cal l o ad D eci s i o ns : L o ad Sh ed d i n g u n d er co n t i n g en cy C o ns tra i nt s : Fi rs t - s t ag e d eci s i o n s O p erat i o n al co n s t rai n t s Con t i n g en ci es u n d er ex t rem e e v en t s A b u n d an ce o f d i s p at ch ab l e res o u rces F i rs t Sta g e: Res i l i en ce Pl an n i n g Objecti v e: , Mi n i m i ze in v es t m en t co s t s D eci s i o ns : L i n e h ard en i n g m eas u res St o rag e s i t i n g & si zi n g C o nstra i nt s : In v es t m en t Bud g et D eci s i o n - d ep en d en t U n cert ai n t y Ri s k - i n f o r m ed C o o rd i n at i o n Seco nd Sta g e: Ri s k - i n f o r m ed o p erat i o n N o rm al Con d i t i o n Objecti v e : Min i m i ze o p erati o n cos t s D eci s i o ns : Mu l t i - res o u r ce p re - d i s p at ch C o ns tra i nt s : O p erat i o n al co n s t rai n t s In fo rm at i o n o f ex t rem e ev en t s Fig. 3. The proposed two-stage risk-informed resilience planning model. off between costs under different scenarios to enhance system resilience while maintaining economic ef ficiency . A. Objective Function The object function of the proposed two-stage stochastic resilience planning model is as follows: min X l ∈ L c l x l + X i ∈ N c b Z i + E s [ Q ( x , z , s )] (14) The first and second terms of (14) are the in vestment cost of line hardening and energy storage, respectiv ely . The third term is the expected v alue under normal and extreme operation, respectiv ely . The mathematical representation of this term is as below: E s [ Q ( x , z , s )] = X s ∈ S p s ( ψ N ( x , z , s ) + ψ E ( x , z , s )) (15) where ψ N ( x , z , s ) = min X t ∈ T N ( X i ∈ N c g P g ,N i,t,s + X i ∈ N c dis P dis,N i,t,s + X i ∈ N c w P wc,N i,t,s + X i ∈ N c N LS P LS,N i,t,s ) (16) ψ E ( x , z , s ) = min X t ∈ T E X i ∈ N c E LS P LS,E i,t,s (17) The three terms in (16) aim to minimize the operation costs of generators, storage, wind curtailment, and load shedding under normal conditions, respectively . Equation (17) is to minimize the cost of load shedding under emer gency conditions, i.e., ice storms. It should be noted that load shedding is introduced in the normal operation stage as a proactiv e measure to mitigate the impact of anticipated extreme ev ents, which are leveraged by predictable information. The rationale is that pre-emptiv e charg- ing of storage may introduce an imbalance between generation and demand. Therefore, pre ventiv e load shedding and storage charging are necessary to deal with potential po wer disruption. Since cost-effecti veness is still the major objectiv e during normal operation, the coef ficient c N LS can be adjusted to achieve the trade-off between economic efficiency and system resilience. B. Constraints In this two-stage stochastic mixed-integer linear programming model, the constraints can be divided into three parts, including 6 in vestment budgets, operational constraints under normal condi- tions and extreme ev ents, respectively . In the first stage, the inv estment decisions are made according to related budgets, which are gi ven as follows: X l ∈ L c l x l ≤ C L (18) X i ∈ N c b Z i ≤ C Z (19) x l ∈ { 0 , 1 } , ∀ l ∈ L (20) In the second stage, there are two types of constraints. One is for normal operation prior to e xtreme ev ents, the other is for emergent operation during ice storms. Since there is an anticipation of extreme ev ents and the anticipatory preparation time given by the predictiv e information is represented by T N , operators are capable of planning preparation methods such as multi-resource re-dispatch. The operational constraints under normal conditions are as follo ws: P g ,N i,t,s + X j i ∈ L P N j i,t,s − X ij ∈ L P N ij,t,s − ( P D,N i,t,s − P LS,N i,t ) + P w,N i,t,s − P wc,N i,t,s + P ch,N i,t,s − P dis,N i,t,s = 0 (21) P N ij,t,s + P N j i,t,s = B ij ( θ i,t,s − θ j,t,s ) (22) − P Max ij ≤ P N ij,t,s + P N j i,t,s ≤ P Max ij (23) P Min g ≤ P g ,N i,t,s ≤ P Max g (24) 0 ≤ P LS,N i,t ≤ α i P D,N i,t,s (25) 0 ≤ P wc,N i,t,s ≤ P w,N i,t,s (26) x dis,N i,s + x ch,N i,s ≤ 1 (27) x dis,N i,s , x ch,N i,s ∈ { 0 , 1 } (28) 0 ≤ P dis,N i,t,s ≤ x dis,N i,s P Max b (29) 0 ≤ P ch,N i,t,s ≤ x ch,N i,s P Max b (30) S N i,t − 1 ,s − S N i,t,s = ( P dis,N i,t,s /η + − P ch,N i,t,s /η − ) · ∆ t (31) 0 ≤ S N i,t,s ≤ Z i (32) P Max b ρ N ≤ Z i (33) where (21) describes the power balance between supply and demand, (22) and (23) represent the power flo w equations and line flo w limits, (24) represents the generator output limits and (25) sets the tolerable load shedding ratio of each bus and the critical loads usually have lower tolerance of load shedding, (26) sets the wind curtailment limits, (27) and (28) describe that the energy storage can not charge and discharge simultaneously , (29) and (30) set the dischar ging/charging limits of ener gy storage, (31) models the dynamic equation of energy storage. (32) sets the state-of-charge limit, and (33) relates the energy and po wer ratings of storage. Similarly , the operational constraints during extreme e vents are as follows: P g ,E i,t + X j i ∈ L P E j i,t,s − X ij ∈ L P E ij,t,s − ( P D,E i,t,s − P LS,E i,t ) + γ i,t,s ( P w,E i,t,s − P wc,E i,t,s ) + P ch,E i,t,s − P dis,E i,t,s = 0 (34) − (1 − µ ij,t,s ) P Max ij ≤ P E ij,t,s + P E j i,t,s ≤ (1 − µ ij,t,s ) P Max ij (35) − µ ij,t,s M ≤ P E ij,t,s + P E j i,t,s − B ij ( θ i,t,s − θ j,t,s ) ≤ µ ij,t,s M (36) − P Max ij ≤ P N ij,t,s + P N j i,t,s ≤ P Max ij (37) P Min g ≤ P g ,E i,t,s ≤ P Max g (38) 0 ≤ P LS,E i,t,s ≤ β i P D,E i,t,s (39) 0 ≤ P wc,E i,t,s ≤ γ w,t,s P w,E i,t,s (40) x dis,E i,t,s + x ch,E i,t,s ≤ 1 (41) x dis,E i,t,s , x ch,E i,t,s ∈ { 0 , 1 } (42) 0 ≤ P ch,E i,t ≤ x ch,E i,t,s P Max b (43) 0 ≤ P dis,E i,t ≤ x dis,E i,t,s P Max b (44) S E i,t − 1 ,s − S E i,t,s = ( P dis,E i,t,s /η + − P ch,E i,t,s /η − ) · ∆ t (45) 0 ≤ S E i,t,s ≤ Z i (46) P Max b ρ E ≤ Z i (47) In the abov e constraints, the binary variables γ i,t,s and µ ij,t,s denote the damage status of wind farms and transmission lines, respectiv ely . If the wind farm i is out-of-service at time interv al t in scenario s , then γ i,t,s = 1 , otherwise γ i,t,s = 0 . Similarly , if line ij is damaged at time interv al t in scenario s , then µ ij,t,s = 1 , otherwise µ ij,t,s = 0 . The occurrence of predictable information can influence the normal operation before extreme events to make preparations. The system states at the beginning of the extreme events, or the end of the normal operation, will be different giv en dif ferent preparation times, which can be denoted as: S E i,t 0 ,s = S N i,t n ,s (48) Remarks: While some research has considered operation under normal and emergenc y conditions, the distinctiv e feature of our model lies in the integration of predicti ve information and the coupling between the two stages. Instead of being treated as independent scenarios, preventi ve normal operation (21)-(33) and emergent operation (34)-(47) collectiv ely form a compre- hensiv e scenario through the e xplicit coupling of (48) and the implicit coupling across consecutiv e time intervals. Therefore, our model emphasizes the coordination both prior to and during contingencies, influenced by the anticipatory preparation time, which is determined by predicti ve information and its associated time-varying uncertainty . C. Scenario Generation In general, the sample space of the stochastic programming problem described in (14)-(48) is of infinite dimensions. T o cope with this dif ficulty , we consider finite realizations of scenarios to construct an approximation of the resilience planning problem. The scenario set S is restricted to denote the set of scenarios s with corresponding probabilities p s . The scenario set is generated by applying the general procedure stated in Algorithm 1. D. Solution Algorithm A large number of scenarios result in extremely large-scale MILP resilience planning models. Commonly used stage-wise decomposition methods, such as Branch and Bound, can be com- putationally intensi ve, especially for large-scale problems with binary variables [22]. Benders’ decomposition, which is another 7 Algorithm 1 Scenario Generation 1: Input: Parameters T N , T E and weather-related data. 2: For scenario s = 1 , ..., | S | do 3: Sample ice storm parameters using the related PDFs. 4: Sample κ i,s from N (1 , 0 . 1) , ∀ i ∈ Ω L 5: Generate load profile via (7) 6: For ij ∈ Ω B do 7: Sample T r ij ( s ) via (8) 8: Calculate r ( s ) via (1) 9: Calculate P f ij ( r ) and P f W T ( r ) via (4) and (6) 10: Sample component status µ ij ( s ) and γ w ( s ) 11: µ ij,t /γ w,t ( s ) = ( 1 min { t + T r ij /w ( s ) − 1 , T } ≤ t ≤ T 0 otherwise 12: end for 13: end for general method for solving large-scale mixed-integer stochastic programming problems, requires the con vexity of ψ ( v , s ) and becomes in valid when there are integer v ariables in the second stage. T o mitigate the computational difficulty , the progressi ve hedging algorithm (PHA) is adopted to solve the large-scale RDDIP model. PHA first decomposes the e xtensive form of the original problem according to each scenario. Then, the penalized v ersions of the subproblems will be solv ed parallelly and iterativ ely until conv ergence. T o implement the PHA, the RDDIP model is rewritten in the following compact form: min v c ⊤ v + X s ∈ S p ( s ) ψ ( v , s ) (49) s . t . Av ≤ B (50) v ∈ Z m 1 + × R n 1 − m 1 (51) where v represents the binary line hardening decision vari- ables and the storage siting and sizing decision variables in the first stage, as shown in (14). c is the corresponding cost coefficient vector . (50) is the matrix form of the inv estment budget constraints as shown in (18) - (20). ψ ( v , s ) represents the second-stage operation problem under scenario s , which can be expressed as: ψ ( v , s ) = min d ⊤ ( s ) q (52) s . t . Dq ≤ F ( s ) − H ( s ) v (53) q ∈ Z m 2 + × R n 2 − m 2 (54) where q represents the second-stage decision variables. (53) is the matrix form of the operational constraints corresponding to (21) - (48). T o render the problem solv able, we construct an approxima- tion of the problem by considering finite scenarios | S | . Then, we can obtain the scenario formulation of the RDDIP model: min X s ∈ S [ c ⊤ v ( s ) + p ( s ) d ( s ) ⊤ q ( s )] (55) s . t . Av ≤ B (56) D q ≤ F ( s ) − H ( s ) v (57) v − ˆ v = 0 (58) v , ˆ v ∈ Z m 1 + × R n 1 − m 1 (59) q ∈ Z m 2 + × R n 2 − m 2 (60) Algorithm 2 The Progressiv e Hedging Algorithm for T wo-Stage Stochastic MILP Resilience Planning Problems 1: Initialization: Let k ← 0 and w k ( s ) ← 0 . F or each s ∈ S calculate: ( v k +1 ( s ) , q k +1 ( s )) ∈ argmin ( v , q ) ∈ Ψ( s ) c ⊤ v + d ( s ) ⊤ q Ψ( s ) := { v ∈ Z m 1 + × R n 1 − m 1 , q ∈ Z m 2 + × R n 2 − m 2 : Av ≤ B , D q ≤ F ( s ) − H ( s ) v } 2: While all scenario solutions v k ( s ) are unequal do 3: Iteration Update: k ← k + 1 4: Aggregation: ˆ v k ← P s ∈ S p ( s ) v k 5: Multiplier Update: w k ( s ) ← w k − 1 ( s ) + ρ ( v k ( s ) − ˆ v k ) 6: Decomposition: For each s ∈ S , calculate: ( v k +1 ( s ) , q k +1 ( s )) ∈ argmin ( v , q ) ∈ Ψ( s ) { c ⊤ v + d ( s ) ⊤ q + w k ( s ) ⊤ v + ρ 2 ∥ v − ˆ v k ∥ 2 } 7: end while where ˆ v is the copy of the first-stage variables for each sce- nario. (58) is the so-called non-anticipati vity constraints, which stipulate that in all feasible solutions, the first-stage decisions are independent on scenarios. W ithout (58), the extensiv e form of the scenario formulation can be decomposed by scenario. The PHA is initialized by decomposing the lar ge-scale, two- stage RDDIP model into scenario-specific subproblems, each of which is then solv ed independently . In each iteration, the solution of the individual scenario solutions will be projected onto the subspace of non-anticipative constraints for aggre gation. Non-anticipativity is enforced by using penalties for multiplier updates. Then, each subproblem whose first-stage objecti ves are perturbed by the multipliers will be solved. The procedure of implementing the PHA to solv e the two-stage stochastic MILP resilience planning problem is sho wn in Algorithm 2. V . C A S E S T U D I E S A. Experiment Settings In this paper , we projected the 118-bus transmission systems in T exas, where ice storms occur frequently , for case studies. W e modified those systems by incorporating wind f arms. Specif- ically , four wind farms with a capacity of 500MW are added in bus 23, 70, 94 and 103. The historical data from the ERCO T market during the winter of 2020, a period that witnessed sev ere ice storms in T exas, is used for load and wind po wer scenario generation. T o ensure comparability across cases, we fix the total simulation time T in the second stage as 36 hours within a 1-hour resolution. The contingenc y operation duration T E is 24 hours and the preparation time T N varies from 2 hours to 12 hours respectiv ely to analyze the impact of predictive information. The normal operation without kno wing the occurrence of ice storms should be T 0 = T − T E − T N . The objecti ve function and operational constraints are the same as (21)-(33) except that no load shedding is allo wed during this period. For in vestment parameters, the cost of line hardening is set as 1 million $/mile. Generic community-scale energy storage is assumed and does not have specific restrictions on its placement in the transmission system. The capital costs of the energy storage are set to be 75$/kWh and 1300$/kW and are pro-rated on a daily basis using the net present value approach in [23]. The lifetime of the energy storage is assumed to be 10 years with an annual discount rate of 5%. The energy-to-power ratio is set 8 T ABLE I P L AN NI N G S T R A T E GI E S W I T H D IFF ER EN T C O N S I D ER A T IO NS Strategy Anticipatory Preparation Prediction Uncertainty Decision- Dependency I ✓ ✓ ✓ II × \ ✓ III ✓ × ✓ IV ✓ ✓ × T ABLE II L O A D - S H ED DI N G C OS TS O F D I FF E RE NT S T RAT E G IE S ($ ) Strategy Preparation Emergenc y I 149146.12 8864831.45 II \ \ III 203537.94 8808182.29 IV 193033.51 9126962.12 to be 6 hours and both the charging and discharging ef ficiencies are set as 0.9 [24]. For operational parameters, the penalty costs of load shedding during contingencies and wind curtailment are assumed to be 2000$/MWh and 500$/MWh. The time-varying load-shedding penalty during preparation is assumed to be encapsulated using a monotonically decreasing exponential function: c N LS ( t ) = a bt + c , where a ( a > 1) , b and c are parameters that indicate the confidence lev el in the forecasts. A higher penalty corresponds to greater uncertainty . The impact of varying levels of uncertainty in predicti ve information will be ev aluated in the subsequent section. The load-shedding costs of critical buses with higher priority are 2 times that of non-critical buses. For pre ventiv e load shedding, 10% of the critical buses and 20% of normal buses are allowed. During ice storms, 20% critical load shedding is allowed and the normal load shedding will be flexible since the operational objectiv e will be changed to retain the critical load. The procedure in section III is employed to generate contingency scenarios. The probabilities and intensities of ice storms can be estimated using historical data. All models and algorithms are implemented using Julia. All the stochastic mix ed-integer programming problems are solved using the Gurobi solver on a PC running the 64-bit W indows operating system with AMD Ryzen 9 5900X CPU clocking at 3.7 GHz and 64 GB of RAM. The relativ e con vergence gap has been set to 1%. B. P erformance of the T wo-stage RIDDRP model T o ev aluate the efficiency of the proposed RIDDRP model, which incorporates the information and DDU, we compared the amount of load shedding during ice storms of four resilience enhancement strategies with different considerations, as shown in T able I. Strate gy I is the proposed model. Strategy II sets the preparation time T N as 0, implying no consideration of predictiv e information. Strategy III adopts a constant prev enti ve load-shedding penalty , disregarding the uncertainty in predictive information. Strategy IV ignores the DDU and excludes line- hardening strategies. The results are sho wn in T able II. Under the given storage budget, strategy II fails to meet critical load-shedding limits, resulting in infeasible solutions. This underscores the importance of considering predictive in- formation for prior preparation. Strategy III ignores the time- varying forecasting uncertainty , leads to a substantial increase in prev entive load-shedding. Notably , this increase in cost is sig- nificantly higher than the reduction in emergenc y load-shedding, rendering it an inferior solution. Strategy IV ignores the benefits of line hardening and yields the highest costs in both prev enti ve and emergency load-shedding. This indicates the necessity of considering the ef fect of robust system design to further enhance system resilience. C. Risk-informed Resilience Enhancement Outcomes As mentioned in Section II, predictiv e information can influ- ence risk-informed operations by influencing both the time and dispatchable resources av ailable to operators. In this study , we in vestigate the resilience of the test system under each inv est- ment plan considering div erse uncertainty lev els of predicti ve information, which is parameterized by the preparation time and the pre venti ve load-shedding penalty . Fig. 4 illustrates the load shedding during both the risk-informed preparation phase and ice storms in the test system. Notably , as the preparation time increases, the preventi ve load shedding increases while the load shedding during ice storms decreases. For instance, elev ating the prev entive non-critical load shedding by 33.4 MWh can lead to a substantial reduction of 92.4 MWh in load shedding amidst ice storms. This demonstrates that even a marginal amount of pre ventiv e load shedding can precipitate a multiple times decrease in load shedding during contingencies, as observ ed in the case studies, and potentially e ven higher for lar ger systems, underscoring the imperati ve of incorporating risk-informed op- erations prior to anticipated e vents. Giv en that we map the uncertainty of predicti ve information into time-varying pre ventiv e load-shedding costs, the expected prev entive and contingenc y load-shedding costs under various lev els of predictiv e information uncertainty are presented in Fig. 5. Despite incurring a higher load-shedding penalty prior to anticipated ice storms, the benefits of pre venti ve load-shedding are discernible. A relatively modest increase in advance load- shedding costs can precipitate a more substantial reduction in costs when navigating higher demand during ice storms. Thus, the incorporation of predictive information can enable a more resilient operation in the face of extreme e vents. Furthermore, we in vestigate the impact of the confidence lev el in the predictive information on resilience planning. W e selected fiv e dif ferent load-shedding penalty curves, each represented by an exponential function with v arying decay parameters, to simulate different levels of time-v arying forecasting accurac y . These curves all con ver ge to a penalty of 2000$/MWh, which is the designated load-shedding penalty during ice storms. A faster decay implies lo wer forecasting accuracy and higher uncertainty , thereby increasing the risks associated with performing pre ven- tiv e load-shedding. The results are sho wn in Fig. 6. Similarly , operational costs decrease as the preparation time increases across all uncertainty levels. Ho wever , under a fixed preparation                               Fig. 4. Prev entiv e and contingency load shedding in IEEE-118 b us system. 9                                 Fig. 5. Load shedding costs in IEEE-118 bus system. 1 2 3 4 5 10.04 10. 06 10.08 10.10 10.12 10.14 10. 16 10.18 2 4 6 8 10 12 Op era tiona l Cos ts (Mill ion $) Anti ci pa to ry pre pa rat io n ti m e (Ho ur) Fig. 6. Operational costs under various preparation time and forecasting confidence levels in IEEE 118 system. time, the increase in operational costs due to changes in the confidence lev el is less pronounced. This indicates that the proposed model is less sensitiv e to the variances in the time- varying confidence le vel in this case. D. The V alue of Pr edictive Information As mentioned abo ve, the strate gic utilization of predicti ve information presents a balance between enhancing system re- silience and incurring additional costs due to the time-v arying uncertainty inherent in predictive information. T o quantify the value of predictiv e information, in this section, we first in- vestigate the total operation costs during the entire operational horizon, which maintains consistenc y for all cases. The results are sho wn in Fig. 7. It can be noticed that ev en though the uncertainty lev el of predictive information may increase as the time until the anticipated ice storms lengthens, the total opera- tional costs can still witness a reduction of $105,951 by transi- tioning to risk-informed operation by taking preventi ve actions more proactively . As the anticipatory preparation time lengthens, initiating pre ventiv e load-shedding earlier incurs higher penalty costs. Nonetheless, when juxtaposed with the potential cost benefits during contingencies, accepting slightly elev ated risks to proacti vely mak e preparations can pa ve the way for enhanced system resilience in the face of ice storms. Then, we calculate the marginal cost increments of pre venti ve load shedding associated with e xtending the anticipatory prepa- ration time in the IEEE-118 system, which is sho wn in T able III. It can be noticed that there is a discernible increase in marginal costs with extended preparation times. For instance, the marginal cost for extending preparation from 6 to 8 hours is $12,824.61, while it more than doubles to $27,207.76 for an extension from 8 to 10 hours. The results reveal a critical point where the cost of additional preparation outweighs the benefits. While longer preparation enables more comprehensive measures to                       Fig. 7. T otal operational costs under various anticipatory preparation time. T ABLE III M A RG IN A L P R E V EN T I V E O P E R A T I O NA L C O S T I N C R E M EN T Anticipatory Preparation T ime (Hour) Marginal Costs ($) 2 − 4 9093.14 6 8140.09 8 12824.61 10 27207.76 12 17169.74 mitigate load shedding during emergencies, the associated higher marginal costs underscore the need for a judicious selection of preparation time, balancing the benefits of enhanced resilience against escalating costs. E. Balancing Investment Costs and System Resilience via Risk- informed Planning In this section, we inv estigate the relationship between in- vestment decisions and risk-informed operations influenced by predictiv e information. For line-hardening decisions, as the an- ticipatory preparation time extends from 2 hours to 12 hours, the hardening costs are 2901.96, 2953.91, 2836.15, 2953.91, 2865.97, 2865.97 Milion $ respectiv ely . Initially , the system planner is equipped with enhanced insights into system op- eration, potentially necessitating more line hardening to pre- emptiv ely mitigate load shedding. This can be observed from the results that the line hardening costs increase from 2901.96 million $ to 2987.93 million $ as the anticipatory preparation time extends from 2 to 6 hours. Howe ver , as this time extends to 12 hours, the line hardening costs decrease and con verge to 2865.97 million $. This can be explained by the fact that there are more dispatchable resources av ailable, i.e., energy storage in this study , thereby enhancing the system’ s flexibility in managing anticipated contingencies by achie ving a trade-of f between uncertain information and potential disruptions. Under the circumstances, the system can no w le verage more econom- ically viable alternatives to alleviate the impacts of potential system failures. Thereby , the imperati ve to harden additional lines diminishes. Based on the results, this dynamic interplay between enhanced predictiv e information, resource a vailability , and system flexibility underscores the intricate balance that sys- tem planners and operators must na vigate to optimize inv estment costs while ensuring resilient system operation in the face of anticipated disruptiv e ev ents. The impact of v arying predictive information on energy stor- age planning decisions is illustrated in Fig. 8, which presents some ener gy storage siting and sizing results in the IEEE 118- bus system. While the siting decisions are almost the same 10 83 90 117 118 13 95 21 96 53 41 82 44 0 50 100 150 200 250 300 350 2 4 6 8 10 12 St orage Si ti ng & Si zing (MWh) Antici pator y preparatio n ti me (Hou r) Fig. 8. Illustrative example of siting and sizing results of energy storage in IEEE 118-bus System. among dif ferent cases, the allocation varies significantly gi ven different anticipatory time. For instance, for Bus 96, the sizing of the energy storage evolv es from about 36 MWh to 80 Mwh, rev ealing a significant increase. Con versely , Bus 95 shows a different pattern, with sizing decisions decreasing from about 112 MWh to 65 MWh as risk-informed operation stages increase from 2 to 12 hours. This indicates that by taking into account the influence of the predictiv e information, the system planner can achiev e a more risk-informed allocation of resources, resulting in a synergistic balance between economic and resilient operations in the face of future extreme events. V I . C O N C L U S I O N Recognizing the piv otal role of predicti ve information, this paper proposed a two-stage risk-informed decision-dependent resilience planning (RIDDRP) model against ice storms. The RIDDRP model effecti vely integrates strategic line hardening and ener gy storage planning decisions to achie ve risk-informed operation considering both normal and emergency conditions. The model captures the DDU of contingency distributions in- troduced by line hardening decisions, the inherent uncertainty in predictive information, and the exogenous uncertainties asso- ciated with ice storms. T o alleviate the computational burdens, PHA is employed to solve lar ge-scale mixed-integer program- ming problems. Numerical studies rev eal that the RIDDRP model can enhance system resilience from two aspects. Firstly , it informs in vestment decisions by facilitating a more judicious allocation of resources considering multi-source uncertainties. Secondly , it lev erages the predicti ve information to facilitate risk-informed coordination between normal operations, which can make proactive preparations, and emergenc y operations dur - ing contingencies, by optimizing the utilization of dispatchable resources. Furthermore, the value of predicti ve information is quantified to analyze its impacts on planning and operation. This provides insights into identifying more influential properties for risk-informed operations. A ke y advantage of the proposed model is the incorporation of predictiv e information at the planning stage, which enables system planners to make resilient in vestment decision-making through a more efficient allocation of resources. Consequently , this approach fosters a harmonious balance between economic efficienc y and resilience, equipping power systems to better withstand and recov er from future extreme e vents. R E F E R E N C E S [1] M. Panteli, C. Pickering, S. W ilkinson, R. Dawson, and P . 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