Tempered Christoffel-Weighted Polynomial Chaos Expansion for Resilience-Oriented Uncertainty Quantification

T empered Christof fel-W eighted Polynomial Chaos Expansion for Resilience-Oriented Uncertainty Quantification Mahsa Ebadat-Parast, Xiaozhe W ang Department of Electrical and Computer Engineering , McGill University , Montreal, Quebec, Canada Email: mahsa.ebadatparast@mail.mcgill.ca,xiaozhe.wang2@mcgill.ca Abstract —Accurate and efficient uncertainty quantification is essential f or resilience assessment of modern power systems under high impact and low probability disturbances. Data driven sparse polynomial chaos expansion (DDSPCE) pro vides a computa- tionally efficient surrogate framew ork but may suffer fr om ill conditioned regression and loss of accuracy in the distrib ution tails that determine system risk. This paper studies the impact of regr ession weighting schemes on the stability and tail accuracy of DD-SPCE surrogates by introducing a tempered Christoffel weighted least squares (T -CWLS) formulation that balances numerical stability and tail fidelity . The tempering exponent is treated as a hyperparameter whose influence is examined with respect to distributional accuracy compar ed with Monte Carlo simulations. Case studies on distribution system load shedding show that the proposed method reduces 95th percentile deviation by 16%, 5th per centile deviation by 6%, and impro ves the regr ession stability index by over 130%. The results demonstrate that controlling the weighting intensity directly influences both stability index and the accuracy of tail prediction. Index T erms —Power system resilience, uncertainty quantifi- cation, data driven polynomial chaos expansion, Christoffel weighted regression, surrogate modeling I . I N T R O D U C T I O N Increasing high-impact, low-probability (HILP) e vents such as hurricanes, wildfires, and cyberattacks highlight the need for r esilience in power systems,defined as the ability to anticipate, absorb, adapt, and recover [1], [2]. T o enable resilience- oriented decision-making, researchers have increasingly re- lied on probabilistic modeling and uncertainty quantification framew orks [3]. Among them, Monte Carlo simulation (MCS) remains the most accurate b ut suf fers from prohibiti ve com- putational cost for large-scale systems. Uncertainty in power system resilience has also been modeled through scenario- based and Monte Carlo methods [4], [5]. Rob ust and hybrid stochastic–robust optimization frameworks hav e further been applied to ensure feasible operation under extreme condi- tions [6], [7]. While these methods capture representative scenarios or bounded uncertainty regions, they often entail heavy computational cost and may become ov erly conserv ati ve for large-scale systems, moti v ating the use of surrogate-based uncertainty quantification techniques. Among such methods, This work w as supported by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, NSERC RGPIN-2022-03236, CRC-2023- 00006, and by the Rubin & So Foundation Faculty Scholar A ward. (Corre- sponding author: Xiaozhe W ang.) polynomial chaos expansion (PCE) and its data-driv en variants offer an efficient surrogate modeling framework that approxi- mates the stochastic system response using a limited number of deterministic simulations. Data-driven sparse PCE (DD- SPCE) further eliminates the need for explicit probability distributions, enabling surrogate construction directly from av ailable samples or historical data [8]–[10]. These features make DD-SPCE particularly attracti ve for resilience analysis, where data may be limited and system models are computa- tionally intensiv e to ev aluate. Howe ver , the regression system in DD-SPCE can become ill-conditioned when the experimental design is unev en or the sample set is small, resulting in unstable surrogate coefficients and degraded prediction accuracy . T o improv e stability , Christof fel-weighted least squares (CLS) methods hav e been proposed, assigning in verse lev erage-based weights to reduce the influence of high- response samples [11], [12]. Although this strategy enhances numerical conditioning, it tends to underestimate output tails. Accurately capturing the distribution tail is essential, as it gov erns the probability of extreme disruptions. T o address this trade-off between stability and tail fidelity , this w ork in vestigates ho w dif ferent re gression weighting schemes within a data-driv en sparse PCE framew ork influence the stability of surrogate predictions. In particular , a tempering factor is intro- duced to control the weighting intensity between the ordinary least squares (OLS) and the Christof fel-weighted formulations, enabling a systematic study of how weighting strength affects both numerical conditioning and tail representation. Using MCS results as a benchmark, the study shows that moderate tempering improves the surrogate’ s stability and tail accu- racy under limited data. This balance enables more reliable estimation of resilience-related probabilistic metrics. Overall, adjusting regression weights provides a practical means to control the trade-of f between regression robustness and tail representation in data-driv en polynomial chaos models. I I . R E S I L I E N C E - O R I E N T E D M O N T E C A R L O - B A S E D O P T I M I Z A T I O N M O D E L I N G A. Model Description and Assumptions The system under study comprises three interconnected microgrids (MGs) linked through a common distribution net- work. Under normal conditions, each MG exchanges active power with the upstream grid via its point of common coupling (PCC). Follo wing an extreme ev ent, the distribution network becomes isolated, and the MGs operate in emergency mode ,representing the system’ s resilience response to external dis- turbances. Loads in both the distribution system and MGs are categorized into three priority lev els,from critical to non- critical,based on their importance and service requirements. During emer gencies, MGs should supply critical and high- priority loads, while lo wer -priority ones may be curtailed if local resources are insufficient. System resilience is quantified by the total operation cost which includes a penalty term for unserved load in MGs as an objective function and load shedding in the distrib ution system as a constraint, a widely adopted metric in resilience studies [13]–[15]. Minimizing this cost reflects the system’ s capability to maintain essential ser- vices during disruptions. Primary uncertainties are considered: (i) load demand, and (ii) emer gency duration and (iii) initiation time. A MCS framew ork generates stochastic realizations of these parameters to capture their probabilistic behavior . B. Optimization F ormulation The mixed integer linear programming formulation jointly models normal ( Γ = 0 ) and emer gency ( Γ = 1 ) operation modes, where Γ activ ates emergenc y-only components. The objectiv e function in (1) minimizes the total operating cost, including generation, rene wable energy curtailment, storage usage, and market transactions in normal mode, as well as load shedding penalties under emergency conditions. Microgrid- related constraints (2.a)–(2.l) ensure local activ e power bal- ance, import/export limits, and disconnection logic during iso- lation; enforce unit commitment feasibility and ramping limits for dispatchable units; and model the charging/discharging dynamics and state-of-char ge boundaries of the energy stor - age systems. Distribution-le vel equations (3.a)–(3.c) maintain system-wide active and reacti ve po wer balance, incorporate the linearized DistFlow voltage relations, and preserve power - factor consistency for curtailed loads. Finally , the coupling constraints (4.a)–(4.c) define the allowable inter -microgrid power exchanges through the PCC links and enforce con verter- based coupling limits. Collectively , all constraints and the unified objecti ve yield a comprehensi ve stochastic scheduling model that captures both normal and emergency operation behavior through the resilience control parameter Γ . min " X t ∈ T X m ∈ M X g ∈ G m  c op g P G ω m,g ,t + c su g y m,g ,t + c sd g z m,g ,t  (1) + X t ∈ T X m ∈ M  c ren,m P r en ω m,t + c b,m P dis,ω m,t  + (1 − Γ) X t ∈ T X m ∈ M  π buy t P ω imp,m,t − π sell t P ω exp,m,t  + Γ X t ∈ T X m ∈ M X k ∈ K λ mg k P Lsh ω m,t,k # 0 ≤ P ω imp,m,t ≤ β imp,m,t P max imp , ∀ t ∈ T nr (2.a) 0 ≤ P ω exp,m,t ≤ β exp,m,t P max exp , ∀ t ∈ T nr (2.b) β imp,m,t + β exp,m,t ≤ 1 , ∀ t ∈ T nr (2.c) P ω imp,m,t = P ω exp,m,t = 0 , ∀ t ∈ T em (2.d) P G ω m,g ,t − 1 − P G ω m,g ,t ≤ R RD g x m,g ,t + S D g z m,g ,t (2.e) P G ω m,g ,t − P G ω m,g ,t − 1 ≤ R RU g x m,g ,t − 1 + S U g y m,g ,t (2.f) x m,g ,t P G min m,g ≤ P G ω m,g ,t ≤ x m,g ,t P G max m,g (2.g) y m,g ,t + z m,g ,t ≤ 1 , y m,g ,t − z m,g ,t = x m,g ,t − x m,g ,t − 1 (2.h) S O C ω m,t = S O C ω m,t − 1 + η ch m P ch,ω m,t − P dis,ω m,t η dis m (2.i) S O C min m ≤ S O C ω m,t ≤ S O C max m (2.j) 0 ≤ P ch,ω m,t ≤ I ch,ω m,t P ch, max , 0 ≤ P dis,ω m,t ≤ I dis,ω m,t P dis, max (2.k) I ch,ω m,t + I dis,ω m,t ≤ 1 (2.l) V ω n,t − V ω p,t = R n,p P ω n,p,t + X n,p Q ω n,p,t V 0 (3.a) V min ≤ V ω n,t ≤ V max , | P ω n,p,t | , | Q ω n,p,t | ≤ S max n,p (3.b) P Lsh dis,ω n,t,k Q dis t,n,k = QLsh dis,ω n,t,k D dis t,n,k , ∀ t ∈ T em (3.c) X g ∈ G m P G ω m,g ,t + P dis,ω m,t + P r en ω m,t (4.a) + (1 − Γ)  P ω imp,m,t − P ω exp,m,t  + Γ P ω ent,m,t + Γ X k ∈ K P Lsh ω m,t,k = X k ∈ K D m,t,k + P ch,ω m,t + Γ P ω md,m,t X k X n D dis t,n,k = (1 − Γ)  P ω sub,t + X m η inv ,m P ω exp,m,t − X m P ω imp,m,t η rec,m  (4.b) + Γ  X m η inv ,m P ω md,m,t + X k X n P Lsh dis,ω n,t,k  X k X n Q dis t,n,k = X m Q ω v sc,m,t + Γ X k X n QLsh dis,ω n,t,k (4.c) I I I . P R O P O S E D A P P RO AC H This section examines how regression weighting schemes within a DDSPCE framework influence numerical condi- tioning and predictiv e accuracy . The study compares OLS, Christoffel-weighted least squares, and a temper ed Christoffel- weighted regression, where a scalar e xponent α controls the weighting intensity . The analysis highlights how the weighting strength affects both surrogate stability and the representation of rare events. A Monte Carlo dataset is employed as a reference to ev aluate tail agreement and to guide the selection of α . A. Data-Driven P olynomial Chaos Expansion (DD-PCE) Consider d independent input random variables X = ( X 1 , . . . , X d ) with unkno wn or partially known joint probabil- ity measure. Let { x i } M i =1 denote M realizations of X obtained from simulation or measurement, and let Y = [ Y 1 , . . . , Y M ] ⊤ be the corresponding model ev aluations. a) Moment-based orthogonal polynomial basis.: When analytical probability density functions of the inputs are not av ailable, an orthogonal polynomial basis can be constructed directly from the samples. Follo wing the data-driven frame- work of W ang et al. [9], the univ ariate basis { ϕ k ( x ) } p k =0 is obtained by enforcing the discrete orthogonality condition 1 M M X i =1 ϕ k ( x i ) ϕ ℓ ( x i ) = δ kℓ , k , ℓ = 0 , . . . , p, (5) where δ kℓ is the Kronecker delta. In practice, the coefficients of ϕ k ( x ) are determined by a Stieltjes or Hankel–moment orthogonalization based on the empirical moments µ r = 1 M P M i =1 x r i . For multiv ariate inputs, the polynomial basis functions are constructed by tensorization: ψ ν ( x ) = d Y j =1 ϕ ν j ( x j ) , ν = ( ν 1 , . . . , ν d ) ∈ N d 0 , (6) and a truncation rule (e.g., total degree p ) limits the number of basis terms N = # { ν } . b) Data-driven PCE r epr esentation and OLS formula- tion.: The model response is approximated as Y ( x ) ≈ N X j =1 c j ψ j ( x ) , ψ j ( x ) ≡ ψ ν j ( x ) , (7) where c = [ c 1 , . . . , c N ] ⊤ are unknown coefficients. Let the design matrix Ψ ∈ R M × N hav e entries Ψ ij = ψ j ( x i ) . The coefficients are obtained by solving the ordinary least-squares problem min c J OLS ( c ) = 1 M ∥ Ψ c − Y ∥ 2 2 , (8) whose normal equations read G c = 1 M Ψ ⊤ Y , G = 1 M Ψ ⊤ Ψ . (9) Equation (8) constitutes the or dinary least-squar es r e gression employed in the DD-PCE approach of W ang et al. [10], where all samples are equally weighted. This unweighted formulation serves as the reference for ev aluating how alternative weight- ing strategies influence the surrogate stability and prediction accuracy in the subsequent analysis. B. Christoffel Function and W eighted Least Squares The stability of least-squares polynomial approximation de- pends on the conditioning of the empirical Gram matrix [11]: G = 1 M Ψ ⊤ Ψ , (10) where Ψ ∈ R M × N is the design matrix with entries Ψ ij = ψ j ( x i ) . Cohen et al. [11] introduced the Christoffel function to characterize the lev erage of each sample point as K i =   L − T ψ i   2 2 , G = LL ⊤ , (11) where ψ ⊤ i denotes the i th row of Ψ . The stability of the least- squares solution is guaranteed when M κ log M = O (1) , κ = max i K i , (12) which ensures that the Gram matrix (10) is well-conditioned with high probability . T o reduce the influence of high-le verage samples, Liu et al. [16] and Cohen proposed weighting each sample in versely to its Christoffel v alue, leading to the Christoffel-weighted least-squar es formulation: min c J CLS ( c ) = 1 M M X i =1 w i   Y i − N X j =1 c j ψ j ( x i )   2 , (13) with weights defined as w i = M K − 1 i P M j =1 K − 1 j . (14) The associated weighted normal equations are then giv en by G w c = 1 M Ψ ⊤ WY , G w = 1 M Ψ ⊤ WΨ , (15) where W = diag ( w 1 , . . . , w M ) . This weighting effecti vely regularizes G w tow ard the identity and reduces the coherence κ , improving the stability metric M / ( κ log M ) introduced in (12). In the following analysis, this weighting formulation is used to examine how emphasizing low-le verage samples impacts the conditioning of the regression system. C. Pr oposed T emper ed Christof fel-W eighted Re gression (T - CWLS) a) Motivation.: Although the Christoffel-weighted least- squares (CLS) formulation in (13)–(15) stabilizes the regres- sion system, its fully in verse weighting may over -attenuate high-response samples, leading to underestimation of the re- sponse tails. In power -system applications, accurate recon- struction of the tail region of the output probability distribution (e.g., the 95th–99th percentiles) is critical, as it governs rare- ev ent probabilities and system risk. T o balance stability and tail fidelity , we introduce a temper ed exponent α that controls the degree of Christoffel weighting. b) T emper ed weighting formulation.: The proposed re- gression minimizes min c J T - CWLS ( c ) = 1 M M X i =1 w i ( α )   Y i − N X j =1 c j ψ j ( x i )   2 , (16) where the tempered weights are defined as w i ( α ) = M K α i P M j =1 K α j , K i =   L − T ψ i   2 2 , α ∈ R . (17) When α = 0 , all samples are equally weighted and (16) reduces to the ordinary least-squares (OLS) problem of (8); when α = − 1 , it coincides with the in verse Christof fel weighting of (14). The tempered values ( α ) smoothly trade off conditioning improvement and tail accuracy . The weighted normal equations are G w ( α ) c = 1 M Ψ ⊤ W ( α ) Y , G w ( α ) = 1 M Ψ ⊤ W ( α ) Ψ , (18) where W ( α ) = diag ( w 1 ( α ) , . . . , w M ( α )) . In practice, the implementation in volv es computing the Christoffel values K i , applying the tempered weighting w i ( α ) for se veral candidate α v alues, solving the corresponding weighted regressions, and comparing the resulting surrogate outputs with the MCS reference to assess tail agreement. The analysis inv estigates how varying α influences numerical con- ditioning and tail accurac y , pro viding a balanced perspective between the unweighted OLS and the fully weighted CLS formulations. D. Modeling Assumptions and Experimental Design The study in vestigates how dif ferent regression weighting strategies within the DDSPCE framework affect the prediction accuracy of resilience-related metrics. The stochastic input vector is defined as x = [ P load , T start , T dur ] ⊤ , (19) where P load denotes the 24-hour load variation across the system, T start represents the random starting time of the emergenc y event, and T dur is its uncertain duration. For each realization x i , a deterministic scheduling problem is solved to obtain the corresponding system response Y i = f ( x i ) , (20) where Y i is the total operational cost and unserved-load penalty reflecting the system’ s resilience performance. The training dataset D = { ( x i , Y i ) } M i =1 (21) is used to train and compare the OLS, CLS, and T -CWLS surrogates. The analysis focuses on how the choice of regression weighting influences prediction accurac y and stability , rather than on extending the DDSPCE formulation itself. I V . C A S E S T U DY A. Simulation Setup The test system consists of a distribution network connected to three MGs, each equipped with distributed energy resources and local loads. Under normal operating conditions, the MGs can exchange power with the upstream grid. When an extreme ev ent occurs, the connection to the main grid is interrupted, and the distribution system together with the MGs transitions into an emergency mode. During this emergency condition, the MGs must coordinate to supply the critical loads within the distribution system. A schematic of the modified IEEE 34-bus distribution system with three interconnected MGs is illustrated in Fig. 1. B. Result T able I reports how regression strate gies affect the DDSPCE surrogate’ s ability to reproduce the true model’ s statistics. Deviations are measured relativ e to the MCS output; smaller values in the 5% and 95% columns indicate better accuracy in the lo wer and upper tails, which are central to resilience assessment. The OLR,( α = 0 ), represents the unweighted formulation of DDSPCE. It yields unbiased estimates around the mean but underestimates the 5% and 95% percentiles, indicating limited accuracy in extreme responses. The low stability score Score LR further shows weak conditioning, which makes the estimated coefficients sensitive to training noise. Applying the classical Christoffel weighting ( α = 1 ) improv es conditioning by reducing the influence of high- lev erage samples. Although this enhances stability , it also weakens the contrib ution of tail observations, making the sur- rogate less responsi ve to rare e vents and leaving residual errors in the extreme quantiles. The proposed tempered Christof fel weighting introduces a control parameter α that allows the regression to transition smoothly between the two limiting behaviors of OLR and fully weighted Christoffel regression. This tempered weighting preserves sufficient emphasis on tail samples to recover their statistical contribution, while still improving the conditioning of the Gram matrix, as reflected by Score LR > 1 .In fact, the stability index Score LR increases from 0.59 to v alues abov e 1.4, an improvement of more than 130%. Figures 2a and 2b present the absolute percentage deviations in the estimated 5th and 95th percentiles of the output distribution with respect to the reference case(MC). These two metrics quantify ho w accurately each regression weighting configuration reproduces the lo wer and upper tails of the response.Specifically , the av erage 95th–percentile devi- ation is reduced by approximately 16% and the 5th–percentile deviation by about 6% compared with OLR. These quantitative results confirm that tempering the Christof fel function substan- tially enhances both regression robustness and the accuracy of Fig. 1: Modified IEEE 34-bus distribution system integrated with three MGs T ABLE I: Statistical results for various α values for the distribution system’ s load shedding Case 5% 95% µ σ Score LR OLR − 6 . 77% 1 . 93% 0 . 59% 1 . 03% 0 . 59 α : 0 . 1 6 . 73% 1 . 93% 0 . 6% 0 . 97% 0 . 646 α : 0 . 5 − 6 . 5% 1 . 88% 0 . 6% 0 . 64% 0 . 905 α : 0 . 8 − 6 . 42% 1 . 78% 0 . 65% 0 . 36% 1 . 187 α : 1 . 0 − 6 . 5% 1 . 58% 0 . 69% 0 . 24% 1 . 47 α : 1 . 2 − 6 . 33% 1 . 45% 0 . 74% 0 . 12% 1 . 607 α : 1 . 5 − 6 . 05% 1 . 38% 0 . 73% 0 . 54% 1 . 376 α : 2 . 0 − 5 . 9% 1 . 26% 0 . 83% 0 . 85% 0 . 993 (a) (b) Fig. 2: Comparison of absolute percent deviations in the (a) 5th and (b) 95th percentiles of the surrogate-predicted output distributions for different weighting configurations. (a) (b) Fig. 3: (a)Absolute percent deviation in the 95th percentile ( P95% ) for different α values, (b)V ariation of Score LR with tempering exponent α . tail predictions. Therefore, the proposed T –CWLS formulation provides a numerically stable and tail–sensitiv e surrogate that reproduces the extreme behavior of the true MCS model more faithfully than either the unweighted OLR or the con ventional Christoffel–weighted approach. Furthermore, the influence of the tempering parameter α on the objective function of the MGs’ operation is illustrated in Figure 3. Figure 3a shows that the absolute de viation in the 95th percentile decreases as α increases, reaching its minimum near α = 2 , which indicates that stronger tempering enhances the surrogate’ s ability to capture the upper-tail behavior of the distribution. As illustrated in Figure 3b, the minimum acceptable stability threshold, shown by the yellow reference line, marks the lower bound for reliable regression performance. The proposed tem- pered Christoffel approach maintains Score LR values abov e this threshold in some α , whereas the unweighted OLR case falls belo w it. This demonstrates that the proposed weighting not only yields smaller tail deviations but also maintains stable behavior across α values. V . C O N C L U S I O N This paper studied the impact of regression weighting intensity on the stability and tail prediction of DD–SPCE surrogates for resilience assessment. A tempered Christoffel weighted least squares formulation was examined, where the tempering exponent acts as a hyperparameter controlling the weighting applied during regression. Case studies confirm that controlling the weighting intensity directly influences both numerical rob ustness and tail-prediction accuracy , supporting more reliable surrogate-based resilience assessment. 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