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Data-Driven Electricity Price Risk Assessment for Spot Market

Data-Driven Electricity Price Risk Assessment for Spot Market Hindawi International Transactions on Electrical Energy Systems Volume 2022, Article ID 9453879, 11 pages https://doi.org/10.1155/2022/9453879 Research Article 1 1 1 1 2 En Lu , Ning Wang , Wei Zheng , Xuanding Wang , Xingyu Lei , 2 2 Zhengchun Zhu , and Zhaoyu Gong Guangdong Electric Power Trading Center Co., Ltd., Guangzhou 510080, Guangdong Province, China Beijing Tsintergy Technology Co., Ltd., Haidian District, Beijing 100084, China Correspondence should be addressed to Xingyu Lei; lxylxy7@163.com Received 7 October 2021; Accepted 23 November 2021; Published 31 January 2022 Academic Editor: Qiuye Sun Copyright © 2022 En Lu et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Electricity price risk assessment (EPRA) is essential for spot market analysis and operation. *e statistical moments (i.e., the mean and standard deviation) of the price need to be assessed to support market risk control. *is paper proposes a data-driven approach for EPRA based on the Gaussian process (GP) framework. Compared with the deep learning algorithms, GP has two merits: (1) the scale of training sample required is small and (2) the time-consuming hyperparameter tuning process is avoided. However, the direct application of GP for EPRA is not tractable due to the complicated discrete relationship between the system operating status and the electricity price. To deal with that, a data-driven EPRA framework is developed that contains a GP surrogate model for the direct current optimal power flow (DC-OPF) problem and a hybrid model-data-based hybrid electricity price calculation method. To guarantee the accuracy of EPRA, an adaptability criterion and a second verification process based on the Karush–Kuhn–Tucker (KKT) condition are developed to distinguish the samples with GP learning errors. Numerical results carried out on IEEE benchmark systems demonstrate that the proposed method can achieve exactly the same EPRA results as Monte Carlo (MC) simulation, which significantly improved the computational efficiency. efficient assessment method is the basis for spot market 1. Introduction operation and risk control. Current studies focus on the risk To reduce pollution and greenhouse gas emissions, a high caused by the electricity price fluctuation for the risk as- share of renewable energy integration has become one of the sessment in electricity markets. Reference [9] analyses the basic characteristics of the smart grid [1–3]. With the de- optimal electricity procurement problem for large con- velopment of renewable energy and the adoption of loca- sumers considering the electricity price fluctuation. Refer- tional marginal pricing (LMP) methodology, the spot ence [10] proposes a value-at-risk (VaR) and conditional market is full of uncertainties, such as load deviation and VaR (CVaR) assessment for electricity price risk based on renewable variation [4]. *e abovementioned uncertainties historical data. Reference [4] uses the Monte Carlo simu- cause the electricity price to fluctuate violently, bringing lation method in electricity price risk management. Refer- significant operational and planning risks for electricity ence [11] analyses the price risk of power portfolios in market participants. multimarkets based on the well-established mean-variance Risk assessment can provide power system operators model. with priorknowledge and theoreticalbasistoensure safeand For EPRA, the expectation and standard deviation of stable power system operation [5–7]. In the spot market, LMP need to be assessed to support market risk control of electricity price risk assessment (EPRA) is crucial for in- independent system operators (ISOs) [12]. Generally, LMP dependent system operators (ISOs) and market participants. can be obtained based on the direct current optimal power However, it is more volatile and challenging to predict the flow (DC-OPF) model, which is derived from the La- fluctuations in electricity prices than the uncertainties of grangian multipliers of the power balance constraint and power production and consumption [8]. *e reliable and transmission constraints, including an energy component 2 International Transactions on Electrical Energy Systems *e objective of EPRA is to obtain the statistical mo- and a congestion component [13]. Probabilistic optimal power flow (POPF) is able to comprehensively consider ments of the LMP according to various uncertainties of the system operating status. Data-driven methods can build a various uncertainties in the spot market and thus has be- come an effective tool to estimate LMP in the deregulated surrogate model with cheap computation cost to replace the market [14, 15]. time-consuming LMP calculation process. Note that an To solve the POPF problem, two main calculation ap- efficient data-driven EPRA algorithm needs not only high proaches have been developed, namely, model-based and precision and fast computing speed but also good gener- data-based approaches. *e model-based methods can be alization capability with limited training sample, which roughly divided into analytical methods and simulation makes the unique characteristics of GP (e.g., fast training, less intervention, and small sample requirement) an ideal methods. Typical analytical methods, such as the point es- timation method, construct representative samples candidate. However, unlike the POPF problem, the rela- tionship between the input (the system operating status) and according to the probability density function (PDF) of the uncertainty variables [16, 17]. EPRA results can be obtained the output (the LMP) is rather complex due to the dis- continuous property of LMP. Hence, direct learning LMP according to the OPF solutions of representative samples, which is computationally efficient, but complicated math- using data-driven methods is intractable, which will be ematical derivations and strict assumptions are required. shown in the simulation results. Fortunately, the physical Typical simulation methods such as Monte Carlo (MC) model of DC-OPF is known, and this motivates us to de- simulations obtain EPRA results by using massive random velopa newframeworktoachieve theLMP assessment based generated samples that are carried out on the OPF model, on the POPF results by including its physical characteristics. which is reliable but computationally demanding [18, 19]. To this end, a data-driven EPRA approach is proposed based on both physical models and historical data. Com- Recently, data-based machine learning methods have been widely applied in power system [20, 21], showing a prom- pared with existing methods, the proposed data-driven method combines the advantages of the model-based and ising way to achieve EPRA with high precision and fast computational speed. For POPF problem, the core idea of data-based approaches to achieve a more efficient EPRA without accuracy loss. Specifically, we embed the GP sur- the data-based approach is to construct a data-driven sur- rogate model that treats the OPF problem as a functional rogate model for DC-OPF into the model-based EPRA mapping between the system operating status and the OPF process to improve the computational efficiency of the solutions, thus greatly improving the computational effi- traditional model-based method. By providing the strict ciency of the POPF problem. In [22, 23], a deep neural judging criteria (adaptability criterion and a second verifi- network (DNN) approach for solving OPF problems was cation) to determine the inaccurate samples obtained by the developed based on historical data and offline simulations. proposed data-driven method, the accuracy of EPRA is guaranteed. Note that the proposed approach is a general Reference [24] proposed a data-driven machine learning framework for the OPF problem considering the charac- method for EPRA, even in a specific scenario with limited samples. It has the following advantages: (1) the accuracy of teristics of the physical model. However, these data-driven approaches have several technical challenges between the EPRA is maintained. *e EPRA results obtained through POPF problem and EPRA. Several challenges need to be our approach are exactly the same as those of the MC addressed. First, the discrete features of LMP are hard to be method. (2) *e training sample size for learning the LMP learned by the existing data-driven methods. Second, the has significantly reduced thanks to the GP. (3) *e efficiency data-driven methods, such as DNN-based approaches, of EPRA is improved because a large proportion of the time- usually require massive training samples, which may not consuming POPF process is replaced by direct GP mapping. align with the current spot market practice. *ird, the in- *e main contributions of this paper are summarized as herent learning error of the data-driven methods is inevi- follows: table and may yield an unreliable EPRA result. To overcome (1) A data-driven framework is proposed to reduce the the challenges mentioned above, our work combines the scale and accelerate the computational speed of the advantages of the two aforementioned approaches to de- EPRA problem. Specifically, to avoid directly velop a data-driven assisted electricity price risk assessment learning the LMP, a GP surrogate model for the DC- method based on the Gaussian process (GP) and the physical OPF problem is developed to identify key infor- model of DC-OPF. mation for LMP calculation (e.g., the marginal Compared with traditional DNN-based methods generators and congested transmission lines). *en, [25, 26], the GP is a novel machine learning technology that a model-data hybrid EPRA method is proposed by requires smaller training samples and fewer hyper- solving a set of linear equations. *e proposed parameters for learning [27, 28], making the GP align well method can significantly improve the efficiency of with current industry practice. *e GP is used extensively the EPRA without compromising its accuracy. as a nonparametric regression tool in various scenarios, (2) Under this framework, a model-based adaptability e.g., active learning [29], multitask learning [30, 31], criterion and a second verification for EPRA are manifold learning [32], and optimization [33]. However, developed to determine inaccurate samples. Before the learning error of GP is also inevitable. Further advanced technology is required to accurately learn the features of using the sample with marginal generators and congested transmission lines identified by the GP to LMP. International Transactions on Electrical Energy Systems 3 calculate the LMP, physical model information is generator output and demand quantity, respectively. Note used to distinguish the samples with learning errors. that renewables are treated as negative loads in this paper, Hence, the accuracy of EPRA is guaranteed. which are included in D . *e linear objective function of the DC-OPF model is *e rest of the paper is organized as follows: the data- designed to minimize the operating costs associated with driven EPRA framework is developed in Section 2. Section 3 supplying real power to meet the demand requirement. presents the proposed GP surrogate model for DC-OPF. Equation (2) is the system power balance equation, and λ is Numerical results are analyzed in Section 4, and finally, the corresponding Lagrangian multiplier. *e constraints in Section 5 concludes the paper. max (3) and (4) limit the transmission line power flow, and η min and η are the Lagrangian multipliers of the upper and 2. The Data-Driven Framework for EPRA lower transmission limit constraints, respectively. *e constraints in (5) are the operational limits for the real max min In the spot market, the LMP arises from an economic generator power, and ξ , ξ are the Lagrangian multi- i i dispatch. Specifically, the system operator solves a DC-OPF pliers of the upper and lower limits of the generator output problem for the optimal economic generation that meets the constraints, respectively. variational load and renewable energy while satisfies the generationand transmissionconstraints [34]. In fact, there is 2.2. Deduction for the LMP Formulation. To understand the a linear relationship between the LMP and the Lagrangian internal relationship between the LMP and DC-OPF multiplier of the DC-OPF model. *e relationship relies on problem, the KKTcondition is used to analyze the properties the marginal generator and congested transmission line, of LMP. which can be obtained through the DC-OPF solutions. Hence, the key idea of the proposed data-driven framework is to build a GP surrogate model for the DC-OPF problem to 2.2.1. 0e LMP Formulation. According to the KKT con- identify the marginal generator and congested transmission dition, we derive the relationships among the LMP, the line, thus improving the computational efficiency of EPRA. Lagrangian multiplier of the power balance λ, and the dual *e physical characteristics of the DC-OPF model are max min multiplier of the transmission line limits μ and μ . Note l l considered to ensure accuracy. In this section, the LMP that in the following analysis, the saddle point used by the formulation is first studied using a general DC-OPF for- KKT condition corresponds to the global optimum of the mulation and its Karush–Kuhn–Tucker (KKT) condition. OPF model. *en, the data-driven approach is proposed for EPRA. To obtain the LMP for the EPRA, the Lagrangian Note that this paper is focused on the LMP risk arised function of the DC-OPF models (1)–(5) is denoted by LF, as from the uncertainty of load and renewable energy. Within follows: the proposed scope, we assume the topology of power grid is L � 􏽘 c P − λ 􏽘 P − D􏼁 invariable. *e uncertainties from equipment fault are i i i i i∈I i∈I ignored. G min ⎝ ⎠ ⎛ ⎞ − 􏽘 η 􏽘 PT DF × P − D􏼁 + F li i i l 2.1. A General DC-OPF Formulation. A general DC-OPF i∈I l∈I (6) model for economic dispatch is formulated as follows: Objective function: max ⎛ ⎝ ⎞ ⎠ − 􏽘 η − 􏽘 PT DF × P − D􏼁 + F l li i i l l∈I i∈I min 􏽘 L c P . i i (1) i∈I min max − 􏽘 ξ 􏼐P − P 􏼑 − 􏽘 ξ 􏼐P − P 􏼑. i min ,i max ,i i i i i∈I i∈I G G Constraints: *en, the LMP for the load at Bus i is derived from the 􏽘 P − 􏽘 D � 0: λ, i i (2) Lagrangian function in (6) as i∈I i∈I G D zL min max max LMP � � λ + 􏽘 PT DF 􏼐η − η 􏼑. (7) i li l l 􏽘 PT DF × P − D􏼁 ≤ F : η , li i i l l zD (3) l∈L i∈I From (7), we note that the LMP is obtained by the min − 􏽘 PT DF × P − D􏼁 ≤ F : η , li i i l l marginal generator through λ and the congested trans- (4) i∈I max min mission line through η and η . *e relationship be- l l tween the marginal generators and congested transmission min max P ≤ P ≤ P : ξ , ξ , (5) min ,i i max ,i i i lines is discussed in the next section. where c is the generator cost for production; PTDF rep- i li resents the power transfer distribution factor of Bus i to Line 2.2.2. 0e Relationship between Marginal Generators and l; 􏽐 PT DF × (P − D ) is the transmission power flow of Congested Transmission Lines. Based on the KKTcondition, i∈I li i i Line l, which is denoted as PF ; and P and D are the the following equation for generator i can be obtained: l i i 4 International Transactions on Electrical Energy Systems min max zL where η � η − η and I and N are the set and the MG MG min max min max l l l � c − λ − 􏽘 PT DF 􏼐η − η 􏼑 − 􏼐ξ − ξ 􏼑 � 0. i li l l i i (8) number of marginal generators, respectively. *en, the zP l∈I matrix form of the equations above is analyzed. *e matrix max min form of (9) is For the marginal generators, ξ � ξ � 0, and i i equation (8) can be expressed as λ + 􏽘 PT DF η � c , i ∈ I , li l i MG (9) l∈I 1 PT DF · · · PT DF c ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ 11 L1 ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ η ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⋮ ⋮ ⋱ ⋮ ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ ⎥ × ⎢ ⎥ � ⎢ ⎥ . (10) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 1 PT DF . . . PT DF c L1 LN N MG N ×(L+1) MG N ×1 MG MG L (L+1)×1 *ere are N equations and (L+1) unknown variables Hence, the number of congested transmission lines N MG CL in (10), which cannot yield a unique solution. To solve these is equations, (L+1 − N ) variables should be determined in MG N � L − L + 1 − N 􏼁 � N − 1. (11) CL MG MG advance. Hence, the information of the transmission line constraints is introduced. Note that for transmission lines *en, the equations in (10) can be rewritten as without congestion, η � 0. 1 PT DF · · · PT DF ⎡ ⎢ ⎤ ⎥ c 11 L1 ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ η ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢⋮ ⋮ ⋱ ⋮ ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ ⎥ × ⎢ ⎥ � ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (12) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 PT DF . . . PT DF c N 1 N N N CL CL MG N ×N MG N ×1 MG MG MG CL N ×1 MG Based on the above discussion, we know that if the Step 1: Marginal Generators and Congested Transmis- marginal generators and the congested transmission lines sion Line Identification. *e power system operating are known, the LMP can be calculated by solving a set of data (e.g., D ) are collected from historical data or by linear equations in (12). Hence, the identification of mar- running a Monte Carlo simulation, and they are input ginal generators and congested transmission lines is the key into the trained GP surrogate model for DC-OPF. For problem for EPRA. each sample, the DC-OPF outputs (e.g., the generator output and transmission power flow) are immediately obtained. *en, the marginal generators and congested 2.3. 0e Proposed Framework. In this section, a trustworthy transmission lines can be identified with high precision data-driven framework is proposed for EPRA. Based on and fast computation. historical data, a GP surrogate model is developed for DC- Step 2: Distinguishing the Error Samples. According OPF to identify the marginal generators and congested to the discussion presented in Section 2.2, for the transmission lines, which will be introduced in the next DC-OPF problem, we note that the number of mar- section. After identification, the LMP can be obtained im- ginal generators N is n+1 if the number of con- MG mediately without solving the time-consuming DC-OPF gested transmission lines N is n. *is natural CL problem. Unfortunately, inherited learning error in data- property motivates us to adapt it to the proposed driven methods, including the proposed GP surrogate model, framework to distinguish the samples with learning is unavoidable, making the identification of marginal gen- errors. Hence, for each sample with marginal gener- erators and congested transmission lines unreliable. To ators and congested transmission lines identified by overcomethischallenge,anadaptabilitycriterionisdeveloped the GP surrogate model, the proposed adaptability based on the KKTcondition of the physical model discussed criterion is as follows: in Section 2.2. *e proposed framework for EPRA is illus- N + 1 � N . (13) trated in Figure 1. *e three steps are described as follows: MG CL International Transactions on Electrical Energy Systems 5 Step 3 EPRA results Solving linear Solving Step 1 equations DC-OPF Power system operating data Samples Samples with without learning error GP surrogate learning error model Meet Else Samples with Second verification identified MG and CL Adaptability criterion Step 2 Figure 1: Flowchart of the proposed data-driven framework. It should be noted that the proposed adaptability cri- 3. GP Surrogate Model for DC-OPF terion is not strictly complete. *ere are a very few *is section first briefly introduces the GP. *en, the GP samples that meet the adaptability criterion but have surrogate model for DC-OPF is proposed to identify the learning errors. For these samples, the LMP error of all marginal generators and the congested transmission lines. the buses can diverge significantly from the real value *e basic idea of the proposed GP surrogate model is to use because the marginal cost of generation is changed. theGPtoreplacethetime-consumingoptimizationprocessof Hence, a second verification process is proposed based DC-OPF, as illustrated in Figure 2. *e complicated DC-OPF on historical data, as follows: 􏼌 􏼌 features can be represented by the mapping relationship f: D 􏼌 􏼌 􏼌 􏼌 LMP − mean LMP 􏼌 􏼁 􏼌 i i ⟶ P , PF, which is the learning target of the GP. i l (14) ≥ p, mean LMP where mean(LMP ) is the mean value of LMP at Bus i, 3.1. A Brief Introduction to the GP Regression Method. which is obtained from historical data. In this work, we *e GP is generally used to solve hard regression and set p as 50%. classification problems. It is attractive because of its flexible nonparametric nature and computational simplicity. In Step 3: LMP Calculation and EPRA. Based on equation nonparametric statistics, the regularity of a relationship can (13), if the sample meets the adaptability criterion, the be postulated without requiring the dataset to be focused on LMP can be calculated by solving a set of linear an easily describable class. *is efficient property allows the equations; otherwise, we should run DC-OPF to obtain GP to predict the functional behavior inside and outside of the LMP. *en, with all the LMP data in hand, the the input domain with a small sample size [35]. EPRA can be completed by performing statistical *e GP is introduced for regression in this paper. We analysis on the LMP data. Note that the EPRA results denote the regression function by f(·), which is the output of are obtained by the Monte Carlo (MC) simulation, the GP surrogate model. Its corresponding input vector of p which generates massive random samples that are dimensions is denoted as x. For a GP regression problem, a carried out on the OPF model. *e proposed method finite collection of training sample inputs x is denoted as [x , provides an effective tool to replace the time-con- 1 x ,. . ., x ]. Accordingly, the corresponding output f(x) can suming OPF calculation process to reduce the com- 2 n be denoted as [f(x ), f(x ),. . ., f(x )]. According to [36], the putationally demanding of the MC simulation. Hence, 1 2 n model output f(x) is expected to follow a joint multivariate the convergence of the proposed method is the same as normal probability distribution, as follows: the traditional MC. 6 International Transactions on Electrical Energy Systems set (Y, X) and test sample input set X . It follows a Gaussian D D i i distribution N(μ(X ), Σ(X )). *e expected value of Y can be t t t expressed as follows: −1 (20) μ X􏼁 � m X􏼁 + C C (Y − m(X)). t t 21 min c P i i i∊I *us far, the general form of GP regression has been P D i i = 0 i∊I i∊I derived. Equation (20) can be used as a surrogate model for a G G GP surrogate complicated DC-OPF model with a low computational cost. PTDF × (P – P ) ≤ F li i i l model i∊I Further details about the GP can be found in [35, 36]. PTDF × (P – P ) ≤ F li i i l i∊I P ≤ P ≤ P min,i i min,i 3.2. Proposed GP Surrogate Model for DC-OPF. *e basic DC-OPF model introduced in Section 2 can be further expressed as the following linear programming (LP) problem: P , PF P , PF i l i l minc(y) Figure 2: Relationship between DC-OPF and the GP. (21) s.t. Ax + By ≥ b, y ∈ Ω, f x 􏼁 m x 􏼁 C x , x 􏼁 · · · C x , x 􏼁 where y is the output set, including the generation output 1 1 1 1 1 n ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎛ ⎜⎢ ⎥ ⎢ ⎥⎞ ⎟ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ and transmission line power flow, which are the key vari- ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⋮ ⎥ ∼ N⎜⎢ ⋮ ⎥, ⎢ ⋮ ⋱ ⋮ ⎥⎟, (15) ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎝⎢ ⎥ ⎢ ⎥⎠ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ables for determining the marginal generators and congested f x 􏼁 m x 􏼁 C x , x 􏼁 · · · C x , x 􏼁 n n n 1 n n transmission lines, and c(·) is the associated cost function. A and B are the corresponding matrices concerning vectors x where m(·) represents the mean function and C(·,·) is a and y, respectively. With a random x, the DC-OPF in (21) kernel function representing the covariance function. *en, can be cast as a POPF problem. (15) can be rewritten as In the proposed approach, a GP surrogate model is f (X)|X ∼ N(m(X), C(X, X)), (16) developed for the DC-OPF problem with a lower compu- tational cost to improve the effectiveness of POPF. To this where X is an n×p matrix denoted by [x , x ,. . ., x ] . *en, 1 2 n end, the steps for the proposed GP surrogate method are considering that there is independent, identically distributed illustrated below. noise in the model output f(X), the realizations Y can be formulated as 3.2.1. Training Sample Generation. To construct the GP Y|X ∼ N m(X), C(X, X) + σ I , (17) 􏼐 􏼑 surrogate model described in (20) for the DC-OPF problem, the training sample set D � (X, Y) can be obtained from where σ and I are the variances of the noise and an historical operational ISO data or by running an MC sim- n-dimensional identity matrix, respectively. Note that the ulation where the uncertainty vector ω is sampled and the noise is always accounted for in practical implementation in resulting DC-OPF output is calculated for a large number of the GP output. samples. In particular, the Latin hypercube sampling To infer the GP regression output with noise from the methodis implementeddue tothesmall samplerequirement abovementioned sample set (X, Y), a Bayesian inference of the GP. Here, each row of X is an I-dimensional uncertain framework is introduced. It is well known that a Bayesian input vector, including the load demands D of all buses. *e posterior distribution of the model output can be inferred output matrix Y contains columns of I + I output variables from a Bayesian prior distribution of y(x)|x and the likeli- G L as the generator output P and transmission line power flow hoods obtained from the realizations. For a test sample input PF corresponding to each input vector x. set X , the Bayesian prior distribution of Y can be expressed t t as Y |X ∼ N􏼐m X􏼁 , C X , X􏼁 + σ I 􏼑. (18) 3.2.2. GP Surrogate Model Construction. With the training t t t t t n dataset D �[X, Y], we choose the squared exponential (SE) Combined with the training sample set (X, Y), the joint covariance kernel function for our regression problem, i.e., distribution of Y and Y |X can be formulated as follows: x − x 􏼁 x − x 􏼁 2 k ∗ k ∗ C x , x � τ exp − . (22) Y m(X) C C 􏼁 􏼠 􏼡 11 12 SE k ∗ 2l 􏼢 􏼣 ∼ N􏼠􏼢 􏼣, 􏼢 􏼣􏼡, (19) Y |X m X􏼁 C C t t 21 22 *e hyperparameters ξ = (τ, l) can be estimated by the where C � C(X, X)+ σ I ,C � C(X, X ), C � C(X , X), 11 n 12 t 21 t Gaussian maximum likelihood estimator (MLE) method, and C � C(X , X )+ σ I . *en, based on the rules of the 22 t t nt which is optimal under the Gaussian assumption and is easy conditional Gaussian distribution, the Bayesian posterior to implement [31]. Equation (17) with hyperparameters can distribution of Y can be inferred from the training sample t be rewritten as International Transactions on Electrical Energy Systems 7 *e hyperparameter settings of each data-driven method Y|X, ξ ∼ N􏼐m(X), C (X, X) + σ I􏼑. (23) SE are shown in Table 1, which are obtained according to the artificial experience and reference [39]. *e learning error of Based on the MLE, we obtain M1∼M5, which is defined as the average difference between (ξ, σ 􏽢) � argmax log P(Y|X, ξ, σ). the results obtained with M1∼M5 and M0, is evaluated as (24) ξ,σ follows: 􏼌 􏼌 􏼌 􏼌 *e marginal log-likelihood can be expressed as K p 􏼌 􏼌 􏼌 􏼌 􏽐 􏽐 y 􏽢 − y i�1 j�1􏼌 i,j i,j􏼌 􏼌 􏼌 Δ � × 100%, 􏼌 􏼌 K p log P(Y|X, ξ, σ) 􏼌 􏼌 􏼌 􏼌 K · p · 􏽐 􏽐 y 􏼌 􏼌 i�1 j�1 i,j (28) 1 −1 􏼌 T 2 􏼌 􏼌 􏼌 􏼌 � − (Y − m(X)) 􏽨C(X, X|ξ) + σ I 􏽩 (Y − m(X)) y 􏽢 − y n 1 􏼌 􏼌 i,j i,j 􏼌 􏼌 􏼌 􏼌 Δ � 􏽘 􏽘 × 100%, 2 􏼌 􏼌 􏼌 􏼌 K · p y 􏼌 􏼌 i,j i�1 j�1 􏼌 􏼌 n 1 􏼌 2 􏼌 􏼌 􏼌 − log 2 π − log C(X, X|ξ) + σ I . 􏼌 􏼌 2 2 where y 􏽢 and p are the output of the data-driven method i,j (25) and its dimensions; y is the output of Monte Carlo sim- i,j ulation as benchmark; K represents the number of testing After utilizing a gradient-based optimizer, the hyper- samples; and Δ is the MAPE index. parameters are obtained while the GP surrogate model for DC-OPF is fully constructed. 4.1. Evaluation of the Proposed Approach on the IEEE 30-Bus System 3.2.3. 0e Key Information Identification for EPRA. For the output of the proposed GP surrogate model (e.g., P and PF), 4.1.1. Learning Performances of DC-OPF and LMP. We first the learning error is not avoided. To identify the key in- show that the EPRA problem of LMP assessment is more formation for EPRA (e.g., the marginal generator and complicated thanthe OPFproblem,whichcannot belearned congested transmission line) and address the learning error directly by data-driven methods. *e learning performances effect, a relaxing factor ε is introduced. *en, the marginal of the LMP and DC-OPF outputs learned by M1–M4 are generators and congested transmission lines can be iden- compared on IEEE 30 in Table 2. For M1–M3, the number of tified according to the following equations: training samples is set as 10000. For M4 and M5, the number Marginal generator i: of training is set as 200. *e number of testing samples is set as 10000 for all methods. P + ε ≤ P ≤ P − ε . (26) min ,i g,i i max ,i g,i *e results show that directly learning the LMP is in- tractable for data-driven methods because of its discon- Congested transmission line l: tinuous property. Additionally, the DC-OPF problem is 􏼐PF ≥ F − ε orPF ≤ − F + ε 􏼑. (27) more comfortable to learn, making the proposed method l l PF,l l l PF,l based on the learning output of DC-OPF reasonable. After obtaining the marginal generators and congested transmission lines of each sample, based on the framework proposed in Section 2, the EPRA can be achieved in short 4.1.2. Effectiveness of the Proposed Method. To demonstrate order without accuracy loss. the benefits achieved by the proposed approach, we compare the LMP errors of M1–M5 in the IEEE 30-bus system, as shown in Table 3. 4. Numerical Results Several conclusions can be drawn, as follows: In this section, the proposed method is tested on the IEEE (1) Among all the methods for EPRA, the proposed 30-bus system to illustrate its effectiveness, while the IEEE method (M5) achieves the best accuracy, which is 118-bus system is used to demonstrate its scalability to the exactly the same as that of the benchmark method. In larger system. All simulations are performed on a PC the proposed framework, the learning error of the equipped with an AMD Ryzen 5 3600X 6-Core Processor GP is filtered out by the identification process and CPU @ 3.80GHz with 16GB RAM. *e algorithm is model-based adaptability criterion. implemented in MATLAB. (2) Compared with M0, the testing time decreases by *e following methods are compared: 59.34%. *is shows that the computational efficiency (i) M0: Monte Carlo simulation (benchmark) of LMP is significantly improved by the proposed method without accuracy loss. (ii) M1: a neural network method based on SAE [37] (3) Comparing M4 with M1–M3, the results show that (iii) M2: a neural network method based on SDAE [38] the GP can achieve a similar accuracy with a small (iv) M3: a stacked extreme learning machine [24] sample size. However, the learning errors of the data- (v) M4: the Gaussian process [36] driven methods are unavoidable, even with a large (vi) M5: the proposed method number of training samples. 8 International Transactions on Electrical Energy Systems Table 1: Hyperparameter settings of the data-driven methods. Case Method Hyperparameter settings M1 3 layers, 100 nodes per layer, and 200 epochs; learning rate �0.0001, branch size �500 M2 3 layers, 100 nodes per layer, and 200 epochs; learning rate �0.0001, branch size �500 IEEE 30 M3 500 nodes, 50 reduced hidden nodes, and 2 epochs M4 m(x) �0, C(·, ·) �C (·, ·), 100 epochs SE M5 m(x) �0, C(·, ·) �C (·, ·), 100 epochs SE M1 3 layers, 300 nodes per layer, and 300 fine-tuning epochs; learning rate �0.0001; branch size �500 M2 3 layers, 300 nodes per layer, and 300 fine-tuning epochs; learning rate �0.0001; branch size �100 IEEE 118 M3 1000 nodes, 100 reduced hidden nodes, and 4 epochs M4 m(x) �0, C(·, ·) �C (·, ·), 100 epochs SE M5 m(x) �0, C(·, ·) �C (·, ·), 100 epochs SE Table 2: Average error comparison between LMP and DC-OPF on the IEEE 30-bus system Δ of DC-OPF outputs Method Δ of LMP (%) P (%) PF (%) M1 5.98 2.71 5.10 M2 5.93 2.73 5.57 M3 5.95 1.76 2.72 M4 7.11 2.01 3.32 Table 3: LMP average error comparison on the IEEE 30-bus system Method Number of training samples Training time (s) Testing time (s) Δ (%) Δ (%) 1 2 M0 — — 150.01 — — M1 10000 11.72 0.021 6.30 5.98 M2 10000 21.56 0.025 6.41 5.93 M3 10000 1.01 0.34 6.17 5.95 M4 200 7.48 1.38 7.53 7.11 M5 200 10.41 60.99 0 0 Bold values are used for highlighting the results of the proposed method in this paper. 0.4 1 0.3 -1 0.2 -2 -3 0.1 -4 -5 -6 -0.1 -7 -0.2 -8 0 5 10 15 20 25 30 0 5 1015202530 Bus Index Bus Index M1 M4 M1 M4 M2 M5 M2 M5 M3 M3 (a) (b) Figure 3: EPRA error comparison on the IEEE 30-bus system. (a) Error of the mean. (b) Error of the standard deviation. Error of Mean (MW/$) Error of Mean (MW/$) International Transactions on Electrical Energy Systems 9 Table 4: LMP average error comparison on the IEEE 118-bus system. Method Number of training samples Training time (s) Testing time (s) Δ (%) Δ (%) 1 2 M0 — — 227.04 — — M1 10000 94.52 0.07 4.13 3.60 M2 10000 123.51 0.10 3.76 3.43 M3 10000 1.17 0.38 8.39 6.55 M4 200 17.52 6.88 4.30 4.87 M5 200 49.89 123.27 0 0 Bold values are used for highlighting the results of the proposed method in this paper. 0.4 5 0.2 -5 -10 -0.2 -15 -0.4 -20 -0.6 -25 -0.8 -30 -1 -35 -1.2 -40 0 20 40 60 80 100 118 0 20 40 60 80 100 118 Bus Index Bus Index M1 M4 M1 M4 M2 M5 M2 M5 M3 M3 (a) (b) Figure 4: EPRA result comparison on the IEEE 118-bus system. (a) Error of the mean. (b) Error of the standard deviation. (4) For M1–M5, because of the superior quality of the Table 4 and Figure 4, respectively. *e test sample number is GP (e.g., its small sample requirement), the training set to be the same as for the IEEE 30-bus system. *e results show that the proposed method can guar- sample size of the proposed method is much smaller than those of M1–M3, which aligns well with the antee EPRA accuracy even in a large case while improving current industry practice. the computational efficiency. It also shows that the statistical moments (i.e., the mean and standard deviation) diverge far *e errors of the EPRA results are compared, as shown from the real value at Buses 72∼110, which demonstrates in Figure 3. For the proposed method, the EPRA results for that EPRA cannot be achieved by data-driven direct the mean and standard deviation are very accurate. How- learning. ever, for M1∼M4, the estimation results for the mean are relatively accurate, while the standard deviations are far 5. Conclusions and Future Work from the real value obtained by M0, so they cannot achieve anaccurateEPRA. Inparticular, asseen inFigure3,the error *is paper proposes a data-driven framework for EPRA. is abnormally large in some specific buses (e.g., Bus 8 and Specifically, a GP surrogate model is developed to identify Bus 24). *is is because the LMPs at these buses fluctuate the marginal generators and congested transmission lines of much more than other buses, making it challenging to be DC-OPF. *is paves the way for improving the efficiency of directly learned by the data-driven methods, resulting in EPRA. Based on the KKTcondition, an adaptability criterion false information being passed to ISOs and market partic- is proposed to identify samples with learning errors. *e ipants, leading to severe market risks. simulation results show that the proposed method increases EPRA accuracy. Comparisons with recent data-driven 4.2. 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Data-Driven Electricity Price Risk Assessment for Spot Market

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Copyright © 2022 En Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi International Transactions on Electrical Energy Systems Volume 2022, Article ID 9453879, 11 pages https://doi.org/10.1155/2022/9453879 Research Article 1 1 1 1 2 En Lu , Ning Wang , Wei Zheng , Xuanding Wang , Xingyu Lei , 2 2 Zhengchun Zhu , and Zhaoyu Gong Guangdong Electric Power Trading Center Co., Ltd., Guangzhou 510080, Guangdong Province, China Beijing Tsintergy Technology Co., Ltd., Haidian District, Beijing 100084, China Correspondence should be addressed to Xingyu Lei; lxylxy7@163.com Received 7 October 2021; Accepted 23 November 2021; Published 31 January 2022 Academic Editor: Qiuye Sun Copyright © 2022 En Lu et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Electricity price risk assessment (EPRA) is essential for spot market analysis and operation. *e statistical moments (i.e., the mean and standard deviation) of the price need to be assessed to support market risk control. *is paper proposes a data-driven approach for EPRA based on the Gaussian process (GP) framework. Compared with the deep learning algorithms, GP has two merits: (1) the scale of training sample required is small and (2) the time-consuming hyperparameter tuning process is avoided. However, the direct application of GP for EPRA is not tractable due to the complicated discrete relationship between the system operating status and the electricity price. To deal with that, a data-driven EPRA framework is developed that contains a GP surrogate model for the direct current optimal power flow (DC-OPF) problem and a hybrid model-data-based hybrid electricity price calculation method. To guarantee the accuracy of EPRA, an adaptability criterion and a second verification process based on the Karush–Kuhn–Tucker (KKT) condition are developed to distinguish the samples with GP learning errors. Numerical results carried out on IEEE benchmark systems demonstrate that the proposed method can achieve exactly the same EPRA results as Monte Carlo (MC) simulation, which significantly improved the computational efficiency. efficient assessment method is the basis for spot market 1. Introduction operation and risk control. Current studies focus on the risk To reduce pollution and greenhouse gas emissions, a high caused by the electricity price fluctuation for the risk as- share of renewable energy integration has become one of the sessment in electricity markets. Reference [9] analyses the basic characteristics of the smart grid [1–3]. With the de- optimal electricity procurement problem for large con- velopment of renewable energy and the adoption of loca- sumers considering the electricity price fluctuation. Refer- tional marginal pricing (LMP) methodology, the spot ence [10] proposes a value-at-risk (VaR) and conditional market is full of uncertainties, such as load deviation and VaR (CVaR) assessment for electricity price risk based on renewable variation [4]. *e abovementioned uncertainties historical data. Reference [4] uses the Monte Carlo simu- cause the electricity price to fluctuate violently, bringing lation method in electricity price risk management. Refer- significant operational and planning risks for electricity ence [11] analyses the price risk of power portfolios in market participants. multimarkets based on the well-established mean-variance Risk assessment can provide power system operators model. with priorknowledge and theoreticalbasistoensure safeand For EPRA, the expectation and standard deviation of stable power system operation [5–7]. In the spot market, LMP need to be assessed to support market risk control of electricity price risk assessment (EPRA) is crucial for in- independent system operators (ISOs) [12]. Generally, LMP dependent system operators (ISOs) and market participants. can be obtained based on the direct current optimal power However, it is more volatile and challenging to predict the flow (DC-OPF) model, which is derived from the La- fluctuations in electricity prices than the uncertainties of grangian multipliers of the power balance constraint and power production and consumption [8]. *e reliable and transmission constraints, including an energy component 2 International Transactions on Electrical Energy Systems *e objective of EPRA is to obtain the statistical mo- and a congestion component [13]. Probabilistic optimal power flow (POPF) is able to comprehensively consider ments of the LMP according to various uncertainties of the system operating status. Data-driven methods can build a various uncertainties in the spot market and thus has be- come an effective tool to estimate LMP in the deregulated surrogate model with cheap computation cost to replace the market [14, 15]. time-consuming LMP calculation process. Note that an To solve the POPF problem, two main calculation ap- efficient data-driven EPRA algorithm needs not only high proaches have been developed, namely, model-based and precision and fast computing speed but also good gener- data-based approaches. *e model-based methods can be alization capability with limited training sample, which roughly divided into analytical methods and simulation makes the unique characteristics of GP (e.g., fast training, less intervention, and small sample requirement) an ideal methods. Typical analytical methods, such as the point es- timation method, construct representative samples candidate. However, unlike the POPF problem, the rela- tionship between the input (the system operating status) and according to the probability density function (PDF) of the uncertainty variables [16, 17]. EPRA results can be obtained the output (the LMP) is rather complex due to the dis- continuous property of LMP. Hence, direct learning LMP according to the OPF solutions of representative samples, which is computationally efficient, but complicated math- using data-driven methods is intractable, which will be ematical derivations and strict assumptions are required. shown in the simulation results. Fortunately, the physical Typical simulation methods such as Monte Carlo (MC) model of DC-OPF is known, and this motivates us to de- simulations obtain EPRA results by using massive random velopa newframeworktoachieve theLMP assessment based generated samples that are carried out on the OPF model, on the POPF results by including its physical characteristics. which is reliable but computationally demanding [18, 19]. To this end, a data-driven EPRA approach is proposed based on both physical models and historical data. Com- Recently, data-based machine learning methods have been widely applied in power system [20, 21], showing a prom- pared with existing methods, the proposed data-driven method combines the advantages of the model-based and ising way to achieve EPRA with high precision and fast computational speed. For POPF problem, the core idea of data-based approaches to achieve a more efficient EPRA without accuracy loss. Specifically, we embed the GP sur- the data-based approach is to construct a data-driven sur- rogate model that treats the OPF problem as a functional rogate model for DC-OPF into the model-based EPRA mapping between the system operating status and the OPF process to improve the computational efficiency of the solutions, thus greatly improving the computational effi- traditional model-based method. By providing the strict ciency of the POPF problem. In [22, 23], a deep neural judging criteria (adaptability criterion and a second verifi- network (DNN) approach for solving OPF problems was cation) to determine the inaccurate samples obtained by the developed based on historical data and offline simulations. proposed data-driven method, the accuracy of EPRA is guaranteed. Note that the proposed approach is a general Reference [24] proposed a data-driven machine learning framework for the OPF problem considering the charac- method for EPRA, even in a specific scenario with limited samples. It has the following advantages: (1) the accuracy of teristics of the physical model. However, these data-driven approaches have several technical challenges between the EPRA is maintained. *e EPRA results obtained through POPF problem and EPRA. Several challenges need to be our approach are exactly the same as those of the MC addressed. First, the discrete features of LMP are hard to be method. (2) *e training sample size for learning the LMP learned by the existing data-driven methods. Second, the has significantly reduced thanks to the GP. (3) *e efficiency data-driven methods, such as DNN-based approaches, of EPRA is improved because a large proportion of the time- usually require massive training samples, which may not consuming POPF process is replaced by direct GP mapping. align with the current spot market practice. *ird, the in- *e main contributions of this paper are summarized as herent learning error of the data-driven methods is inevi- follows: table and may yield an unreliable EPRA result. To overcome (1) A data-driven framework is proposed to reduce the the challenges mentioned above, our work combines the scale and accelerate the computational speed of the advantages of the two aforementioned approaches to de- EPRA problem. Specifically, to avoid directly velop a data-driven assisted electricity price risk assessment learning the LMP, a GP surrogate model for the DC- method based on the Gaussian process (GP) and the physical OPF problem is developed to identify key infor- model of DC-OPF. mation for LMP calculation (e.g., the marginal Compared with traditional DNN-based methods generators and congested transmission lines). *en, [25, 26], the GP is a novel machine learning technology that a model-data hybrid EPRA method is proposed by requires smaller training samples and fewer hyper- solving a set of linear equations. *e proposed parameters for learning [27, 28], making the GP align well method can significantly improve the efficiency of with current industry practice. *e GP is used extensively the EPRA without compromising its accuracy. as a nonparametric regression tool in various scenarios, (2) Under this framework, a model-based adaptability e.g., active learning [29], multitask learning [30, 31], criterion and a second verification for EPRA are manifold learning [32], and optimization [33]. However, developed to determine inaccurate samples. Before the learning error of GP is also inevitable. Further advanced technology is required to accurately learn the features of using the sample with marginal generators and congested transmission lines identified by the GP to LMP. International Transactions on Electrical Energy Systems 3 calculate the LMP, physical model information is generator output and demand quantity, respectively. Note used to distinguish the samples with learning errors. that renewables are treated as negative loads in this paper, Hence, the accuracy of EPRA is guaranteed. which are included in D . *e linear objective function of the DC-OPF model is *e rest of the paper is organized as follows: the data- designed to minimize the operating costs associated with driven EPRA framework is developed in Section 2. Section 3 supplying real power to meet the demand requirement. presents the proposed GP surrogate model for DC-OPF. Equation (2) is the system power balance equation, and λ is Numerical results are analyzed in Section 4, and finally, the corresponding Lagrangian multiplier. *e constraints in Section 5 concludes the paper. max (3) and (4) limit the transmission line power flow, and η min and η are the Lagrangian multipliers of the upper and 2. The Data-Driven Framework for EPRA lower transmission limit constraints, respectively. *e constraints in (5) are the operational limits for the real max min In the spot market, the LMP arises from an economic generator power, and ξ , ξ are the Lagrangian multi- i i dispatch. Specifically, the system operator solves a DC-OPF pliers of the upper and lower limits of the generator output problem for the optimal economic generation that meets the constraints, respectively. variational load and renewable energy while satisfies the generationand transmissionconstraints [34]. In fact, there is 2.2. Deduction for the LMP Formulation. To understand the a linear relationship between the LMP and the Lagrangian internal relationship between the LMP and DC-OPF multiplier of the DC-OPF model. *e relationship relies on problem, the KKTcondition is used to analyze the properties the marginal generator and congested transmission line, of LMP. which can be obtained through the DC-OPF solutions. Hence, the key idea of the proposed data-driven framework is to build a GP surrogate model for the DC-OPF problem to 2.2.1. 0e LMP Formulation. According to the KKT con- identify the marginal generator and congested transmission dition, we derive the relationships among the LMP, the line, thus improving the computational efficiency of EPRA. Lagrangian multiplier of the power balance λ, and the dual *e physical characteristics of the DC-OPF model are max min multiplier of the transmission line limits μ and μ . Note l l considered to ensure accuracy. In this section, the LMP that in the following analysis, the saddle point used by the formulation is first studied using a general DC-OPF for- KKT condition corresponds to the global optimum of the mulation and its Karush–Kuhn–Tucker (KKT) condition. OPF model. *en, the data-driven approach is proposed for EPRA. To obtain the LMP for the EPRA, the Lagrangian Note that this paper is focused on the LMP risk arised function of the DC-OPF models (1)–(5) is denoted by LF, as from the uncertainty of load and renewable energy. Within follows: the proposed scope, we assume the topology of power grid is L � 􏽘 c P − λ 􏽘 P − D􏼁 invariable. *e uncertainties from equipment fault are i i i i i∈I i∈I ignored. G min ⎝ ⎠ ⎛ ⎞ − 􏽘 η 􏽘 PT DF × P − D􏼁 + F li i i l 2.1. A General DC-OPF Formulation. A general DC-OPF i∈I l∈I (6) model for economic dispatch is formulated as follows: Objective function: max ⎛ ⎝ ⎞ ⎠ − 􏽘 η − 􏽘 PT DF × P − D􏼁 + F l li i i l l∈I i∈I min 􏽘 L c P . i i (1) i∈I min max − 􏽘 ξ 􏼐P − P 􏼑 − 􏽘 ξ 􏼐P − P 􏼑. i min ,i max ,i i i i i∈I i∈I G G Constraints: *en, the LMP for the load at Bus i is derived from the 􏽘 P − 􏽘 D � 0: λ, i i (2) Lagrangian function in (6) as i∈I i∈I G D zL min max max LMP � � λ + 􏽘 PT DF 􏼐η − η 􏼑. (7) i li l l 􏽘 PT DF × P − D􏼁 ≤ F : η , li i i l l zD (3) l∈L i∈I From (7), we note that the LMP is obtained by the min − 􏽘 PT DF × P − D􏼁 ≤ F : η , li i i l l marginal generator through λ and the congested trans- (4) i∈I max min mission line through η and η . *e relationship be- l l tween the marginal generators and congested transmission min max P ≤ P ≤ P : ξ , ξ , (5) min ,i i max ,i i i lines is discussed in the next section. where c is the generator cost for production; PTDF rep- i li resents the power transfer distribution factor of Bus i to Line 2.2.2. 0e Relationship between Marginal Generators and l; 􏽐 PT DF × (P − D ) is the transmission power flow of Congested Transmission Lines. Based on the KKTcondition, i∈I li i i Line l, which is denoted as PF ; and P and D are the the following equation for generator i can be obtained: l i i 4 International Transactions on Electrical Energy Systems min max zL where η � η − η and I and N are the set and the MG MG min max min max l l l � c − λ − 􏽘 PT DF 􏼐η − η 􏼑 − 􏼐ξ − ξ 􏼑 � 0. i li l l i i (8) number of marginal generators, respectively. *en, the zP l∈I matrix form of the equations above is analyzed. *e matrix max min form of (9) is For the marginal generators, ξ � ξ � 0, and i i equation (8) can be expressed as λ + 􏽘 PT DF η � c , i ∈ I , li l i MG (9) l∈I 1 PT DF · · · PT DF c ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ 11 L1 ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ η ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⋮ ⋮ ⋱ ⋮ ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ ⎥ × ⎢ ⎥ � ⎢ ⎥ . (10) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 1 PT DF . . . PT DF c L1 LN N MG N ×(L+1) MG N ×1 MG MG L (L+1)×1 *ere are N equations and (L+1) unknown variables Hence, the number of congested transmission lines N MG CL in (10), which cannot yield a unique solution. To solve these is equations, (L+1 − N ) variables should be determined in MG N � L − L + 1 − N 􏼁 � N − 1. (11) CL MG MG advance. Hence, the information of the transmission line constraints is introduced. Note that for transmission lines *en, the equations in (10) can be rewritten as without congestion, η � 0. 1 PT DF · · · PT DF ⎡ ⎢ ⎤ ⎥ c 11 L1 ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ η ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢⋮ ⋮ ⋱ ⋮ ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ ⎥ × ⎢ ⎥ � ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (12) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 PT DF . . . PT DF c N 1 N N N CL CL MG N ×N MG N ×1 MG MG MG CL N ×1 MG Based on the above discussion, we know that if the Step 1: Marginal Generators and Congested Transmis- marginal generators and the congested transmission lines sion Line Identification. *e power system operating are known, the LMP can be calculated by solving a set of data (e.g., D ) are collected from historical data or by linear equations in (12). Hence, the identification of mar- running a Monte Carlo simulation, and they are input ginal generators and congested transmission lines is the key into the trained GP surrogate model for DC-OPF. For problem for EPRA. each sample, the DC-OPF outputs (e.g., the generator output and transmission power flow) are immediately obtained. *en, the marginal generators and congested 2.3. 0e Proposed Framework. In this section, a trustworthy transmission lines can be identified with high precision data-driven framework is proposed for EPRA. Based on and fast computation. historical data, a GP surrogate model is developed for DC- Step 2: Distinguishing the Error Samples. According OPF to identify the marginal generators and congested to the discussion presented in Section 2.2, for the transmission lines, which will be introduced in the next DC-OPF problem, we note that the number of mar- section. After identification, the LMP can be obtained im- ginal generators N is n+1 if the number of con- MG mediately without solving the time-consuming DC-OPF gested transmission lines N is n. *is natural CL problem. Unfortunately, inherited learning error in data- property motivates us to adapt it to the proposed driven methods, including the proposed GP surrogate model, framework to distinguish the samples with learning is unavoidable, making the identification of marginal gen- errors. Hence, for each sample with marginal gener- erators and congested transmission lines unreliable. To ators and congested transmission lines identified by overcomethischallenge,anadaptabilitycriterionisdeveloped the GP surrogate model, the proposed adaptability based on the KKTcondition of the physical model discussed criterion is as follows: in Section 2.2. *e proposed framework for EPRA is illus- N + 1 � N . (13) trated in Figure 1. *e three steps are described as follows: MG CL International Transactions on Electrical Energy Systems 5 Step 3 EPRA results Solving linear Solving Step 1 equations DC-OPF Power system operating data Samples Samples with without learning error GP surrogate learning error model Meet Else Samples with Second verification identified MG and CL Adaptability criterion Step 2 Figure 1: Flowchart of the proposed data-driven framework. It should be noted that the proposed adaptability cri- 3. GP Surrogate Model for DC-OPF terion is not strictly complete. *ere are a very few *is section first briefly introduces the GP. *en, the GP samples that meet the adaptability criterion but have surrogate model for DC-OPF is proposed to identify the learning errors. For these samples, the LMP error of all marginal generators and the congested transmission lines. the buses can diverge significantly from the real value *e basic idea of the proposed GP surrogate model is to use because the marginal cost of generation is changed. theGPtoreplacethetime-consumingoptimizationprocessof Hence, a second verification process is proposed based DC-OPF, as illustrated in Figure 2. *e complicated DC-OPF on historical data, as follows: 􏼌 􏼌 features can be represented by the mapping relationship f: D 􏼌 􏼌 􏼌 􏼌 LMP − mean LMP 􏼌 􏼁 􏼌 i i ⟶ P , PF, which is the learning target of the GP. i l (14) ≥ p, mean LMP where mean(LMP ) is the mean value of LMP at Bus i, 3.1. A Brief Introduction to the GP Regression Method. which is obtained from historical data. In this work, we *e GP is generally used to solve hard regression and set p as 50%. classification problems. It is attractive because of its flexible nonparametric nature and computational simplicity. In Step 3: LMP Calculation and EPRA. Based on equation nonparametric statistics, the regularity of a relationship can (13), if the sample meets the adaptability criterion, the be postulated without requiring the dataset to be focused on LMP can be calculated by solving a set of linear an easily describable class. *is efficient property allows the equations; otherwise, we should run DC-OPF to obtain GP to predict the functional behavior inside and outside of the LMP. *en, with all the LMP data in hand, the the input domain with a small sample size [35]. EPRA can be completed by performing statistical *e GP is introduced for regression in this paper. We analysis on the LMP data. Note that the EPRA results denote the regression function by f(·), which is the output of are obtained by the Monte Carlo (MC) simulation, the GP surrogate model. Its corresponding input vector of p which generates massive random samples that are dimensions is denoted as x. For a GP regression problem, a carried out on the OPF model. *e proposed method finite collection of training sample inputs x is denoted as [x , provides an effective tool to replace the time-con- 1 x ,. . ., x ]. Accordingly, the corresponding output f(x) can suming OPF calculation process to reduce the com- 2 n be denoted as [f(x ), f(x ),. . ., f(x )]. According to [36], the putationally demanding of the MC simulation. Hence, 1 2 n model output f(x) is expected to follow a joint multivariate the convergence of the proposed method is the same as normal probability distribution, as follows: the traditional MC. 6 International Transactions on Electrical Energy Systems set (Y, X) and test sample input set X . It follows a Gaussian D D i i distribution N(μ(X ), Σ(X )). *e expected value of Y can be t t t expressed as follows: −1 (20) μ X􏼁 � m X􏼁 + C C (Y − m(X)). t t 21 min c P i i i∊I *us far, the general form of GP regression has been P D i i = 0 i∊I i∊I derived. Equation (20) can be used as a surrogate model for a G G GP surrogate complicated DC-OPF model with a low computational cost. PTDF × (P – P ) ≤ F li i i l model i∊I Further details about the GP can be found in [35, 36]. PTDF × (P – P ) ≤ F li i i l i∊I P ≤ P ≤ P min,i i min,i 3.2. Proposed GP Surrogate Model for DC-OPF. *e basic DC-OPF model introduced in Section 2 can be further expressed as the following linear programming (LP) problem: P , PF P , PF i l i l minc(y) Figure 2: Relationship between DC-OPF and the GP. (21) s.t. Ax + By ≥ b, y ∈ Ω, f x 􏼁 m x 􏼁 C x , x 􏼁 · · · C x , x 􏼁 where y is the output set, including the generation output 1 1 1 1 1 n ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎡ ⎢ ⎤ ⎥ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎛ ⎜⎢ ⎥ ⎢ ⎥⎞ ⎟ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ and transmission line power flow, which are the key vari- ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⋮ ⎥ ∼ N⎜⎢ ⋮ ⎥, ⎢ ⋮ ⋱ ⋮ ⎥⎟, (15) ⎢ ⎥ ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ ⎝⎢ ⎥ ⎢ ⎥⎠ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ables for determining the marginal generators and congested f x 􏼁 m x 􏼁 C x , x 􏼁 · · · C x , x 􏼁 n n n 1 n n transmission lines, and c(·) is the associated cost function. A and B are the corresponding matrices concerning vectors x where m(·) represents the mean function and C(·,·) is a and y, respectively. With a random x, the DC-OPF in (21) kernel function representing the covariance function. *en, can be cast as a POPF problem. (15) can be rewritten as In the proposed approach, a GP surrogate model is f (X)|X ∼ N(m(X), C(X, X)), (16) developed for the DC-OPF problem with a lower compu- tational cost to improve the effectiveness of POPF. To this where X is an n×p matrix denoted by [x , x ,. . ., x ] . *en, 1 2 n end, the steps for the proposed GP surrogate method are considering that there is independent, identically distributed illustrated below. noise in the model output f(X), the realizations Y can be formulated as 3.2.1. Training Sample Generation. To construct the GP Y|X ∼ N m(X), C(X, X) + σ I , (17) 􏼐 􏼑 surrogate model described in (20) for the DC-OPF problem, the training sample set D � (X, Y) can be obtained from where σ and I are the variances of the noise and an historical operational ISO data or by running an MC sim- n-dimensional identity matrix, respectively. Note that the ulation where the uncertainty vector ω is sampled and the noise is always accounted for in practical implementation in resulting DC-OPF output is calculated for a large number of the GP output. samples. In particular, the Latin hypercube sampling To infer the GP regression output with noise from the methodis implementeddue tothesmall samplerequirement abovementioned sample set (X, Y), a Bayesian inference of the GP. Here, each row of X is an I-dimensional uncertain framework is introduced. It is well known that a Bayesian input vector, including the load demands D of all buses. *e posterior distribution of the model output can be inferred output matrix Y contains columns of I + I output variables from a Bayesian prior distribution of y(x)|x and the likeli- G L as the generator output P and transmission line power flow hoods obtained from the realizations. For a test sample input PF corresponding to each input vector x. set X , the Bayesian prior distribution of Y can be expressed t t as Y |X ∼ N􏼐m X􏼁 , C X , X􏼁 + σ I 􏼑. (18) 3.2.2. GP Surrogate Model Construction. With the training t t t t t n dataset D �[X, Y], we choose the squared exponential (SE) Combined with the training sample set (X, Y), the joint covariance kernel function for our regression problem, i.e., distribution of Y and Y |X can be formulated as follows: x − x 􏼁 x − x 􏼁 2 k ∗ k ∗ C x , x � τ exp − . (22) Y m(X) C C 􏼁 􏼠 􏼡 11 12 SE k ∗ 2l 􏼢 􏼣 ∼ N􏼠􏼢 􏼣, 􏼢 􏼣􏼡, (19) Y |X m X􏼁 C C t t 21 22 *e hyperparameters ξ = (τ, l) can be estimated by the where C � C(X, X)+ σ I ,C � C(X, X ), C � C(X , X), 11 n 12 t 21 t Gaussian maximum likelihood estimator (MLE) method, and C � C(X , X )+ σ I . *en, based on the rules of the 22 t t nt which is optimal under the Gaussian assumption and is easy conditional Gaussian distribution, the Bayesian posterior to implement [31]. Equation (17) with hyperparameters can distribution of Y can be inferred from the training sample t be rewritten as International Transactions on Electrical Energy Systems 7 *e hyperparameter settings of each data-driven method Y|X, ξ ∼ N􏼐m(X), C (X, X) + σ I􏼑. (23) SE are shown in Table 1, which are obtained according to the artificial experience and reference [39]. *e learning error of Based on the MLE, we obtain M1∼M5, which is defined as the average difference between (ξ, σ 􏽢) � argmax log P(Y|X, ξ, σ). the results obtained with M1∼M5 and M0, is evaluated as (24) ξ,σ follows: 􏼌 􏼌 􏼌 􏼌 *e marginal log-likelihood can be expressed as K p 􏼌 􏼌 􏼌 􏼌 􏽐 􏽐 y 􏽢 − y i�1 j�1􏼌 i,j i,j􏼌 􏼌 􏼌 Δ � × 100%, 􏼌 􏼌 K p log P(Y|X, ξ, σ) 􏼌 􏼌 􏼌 􏼌 K · p · 􏽐 􏽐 y 􏼌 􏼌 i�1 j�1 i,j (28) 1 −1 􏼌 T 2 􏼌 􏼌 􏼌 􏼌 � − (Y − m(X)) 􏽨C(X, X|ξ) + σ I 􏽩 (Y − m(X)) y 􏽢 − y n 1 􏼌 􏼌 i,j i,j 􏼌 􏼌 􏼌 􏼌 Δ � 􏽘 􏽘 × 100%, 2 􏼌 􏼌 􏼌 􏼌 K · p y 􏼌 􏼌 i,j i�1 j�1 􏼌 􏼌 n 1 􏼌 2 􏼌 􏼌 􏼌 − log 2 π − log C(X, X|ξ) + σ I . 􏼌 􏼌 2 2 where y 􏽢 and p are the output of the data-driven method i,j (25) and its dimensions; y is the output of Monte Carlo sim- i,j ulation as benchmark; K represents the number of testing After utilizing a gradient-based optimizer, the hyper- samples; and Δ is the MAPE index. parameters are obtained while the GP surrogate model for DC-OPF is fully constructed. 4.1. Evaluation of the Proposed Approach on the IEEE 30-Bus System 3.2.3. 0e Key Information Identification for EPRA. For the output of the proposed GP surrogate model (e.g., P and PF), 4.1.1. Learning Performances of DC-OPF and LMP. We first the learning error is not avoided. To identify the key in- show that the EPRA problem of LMP assessment is more formation for EPRA (e.g., the marginal generator and complicated thanthe OPFproblem,whichcannot belearned congested transmission line) and address the learning error directly by data-driven methods. *e learning performances effect, a relaxing factor ε is introduced. *en, the marginal of the LMP and DC-OPF outputs learned by M1–M4 are generators and congested transmission lines can be iden- compared on IEEE 30 in Table 2. For M1–M3, the number of tified according to the following equations: training samples is set as 10000. For M4 and M5, the number Marginal generator i: of training is set as 200. *e number of testing samples is set as 10000 for all methods. P + ε ≤ P ≤ P − ε . (26) min ,i g,i i max ,i g,i *e results show that directly learning the LMP is in- tractable for data-driven methods because of its discon- Congested transmission line l: tinuous property. Additionally, the DC-OPF problem is 􏼐PF ≥ F − ε orPF ≤ − F + ε 􏼑. (27) more comfortable to learn, making the proposed method l l PF,l l l PF,l based on the learning output of DC-OPF reasonable. After obtaining the marginal generators and congested transmission lines of each sample, based on the framework proposed in Section 2, the EPRA can be achieved in short 4.1.2. Effectiveness of the Proposed Method. To demonstrate order without accuracy loss. the benefits achieved by the proposed approach, we compare the LMP errors of M1–M5 in the IEEE 30-bus system, as shown in Table 3. 4. Numerical Results Several conclusions can be drawn, as follows: In this section, the proposed method is tested on the IEEE (1) Among all the methods for EPRA, the proposed 30-bus system to illustrate its effectiveness, while the IEEE method (M5) achieves the best accuracy, which is 118-bus system is used to demonstrate its scalability to the exactly the same as that of the benchmark method. In larger system. All simulations are performed on a PC the proposed framework, the learning error of the equipped with an AMD Ryzen 5 3600X 6-Core Processor GP is filtered out by the identification process and CPU @ 3.80GHz with 16GB RAM. *e algorithm is model-based adaptability criterion. implemented in MATLAB. (2) Compared with M0, the testing time decreases by *e following methods are compared: 59.34%. *is shows that the computational efficiency (i) M0: Monte Carlo simulation (benchmark) of LMP is significantly improved by the proposed method without accuracy loss. (ii) M1: a neural network method based on SAE [37] (3) Comparing M4 with M1–M3, the results show that (iii) M2: a neural network method based on SDAE [38] the GP can achieve a similar accuracy with a small (iv) M3: a stacked extreme learning machine [24] sample size. However, the learning errors of the data- (v) M4: the Gaussian process [36] driven methods are unavoidable, even with a large (vi) M5: the proposed method number of training samples. 8 International Transactions on Electrical Energy Systems Table 1: Hyperparameter settings of the data-driven methods. Case Method Hyperparameter settings M1 3 layers, 100 nodes per layer, and 200 epochs; learning rate �0.0001, branch size �500 M2 3 layers, 100 nodes per layer, and 200 epochs; learning rate �0.0001, branch size �500 IEEE 30 M3 500 nodes, 50 reduced hidden nodes, and 2 epochs M4 m(x) �0, C(·, ·) �C (·, ·), 100 epochs SE M5 m(x) �0, C(·, ·) �C (·, ·), 100 epochs SE M1 3 layers, 300 nodes per layer, and 300 fine-tuning epochs; learning rate �0.0001; branch size �500 M2 3 layers, 300 nodes per layer, and 300 fine-tuning epochs; learning rate �0.0001; branch size �100 IEEE 118 M3 1000 nodes, 100 reduced hidden nodes, and 4 epochs M4 m(x) �0, C(·, ·) �C (·, ·), 100 epochs SE M5 m(x) �0, C(·, ·) �C (·, ·), 100 epochs SE Table 2: Average error comparison between LMP and DC-OPF on the IEEE 30-bus system Δ of DC-OPF outputs Method Δ of LMP (%) P (%) PF (%) M1 5.98 2.71 5.10 M2 5.93 2.73 5.57 M3 5.95 1.76 2.72 M4 7.11 2.01 3.32 Table 3: LMP average error comparison on the IEEE 30-bus system Method Number of training samples Training time (s) Testing time (s) Δ (%) Δ (%) 1 2 M0 — — 150.01 — — M1 10000 11.72 0.021 6.30 5.98 M2 10000 21.56 0.025 6.41 5.93 M3 10000 1.01 0.34 6.17 5.95 M4 200 7.48 1.38 7.53 7.11 M5 200 10.41 60.99 0 0 Bold values are used for highlighting the results of the proposed method in this paper. 0.4 1 0.3 -1 0.2 -2 -3 0.1 -4 -5 -6 -0.1 -7 -0.2 -8 0 5 10 15 20 25 30 0 5 1015202530 Bus Index Bus Index M1 M4 M1 M4 M2 M5 M2 M5 M3 M3 (a) (b) Figure 3: EPRA error comparison on the IEEE 30-bus system. (a) Error of the mean. (b) Error of the standard deviation. Error of Mean (MW/$) Error of Mean (MW/$) International Transactions on Electrical Energy Systems 9 Table 4: LMP average error comparison on the IEEE 118-bus system. Method Number of training samples Training time (s) Testing time (s) Δ (%) Δ (%) 1 2 M0 — — 227.04 — — M1 10000 94.52 0.07 4.13 3.60 M2 10000 123.51 0.10 3.76 3.43 M3 10000 1.17 0.38 8.39 6.55 M4 200 17.52 6.88 4.30 4.87 M5 200 49.89 123.27 0 0 Bold values are used for highlighting the results of the proposed method in this paper. 0.4 5 0.2 -5 -10 -0.2 -15 -0.4 -20 -0.6 -25 -0.8 -30 -1 -35 -1.2 -40 0 20 40 60 80 100 118 0 20 40 60 80 100 118 Bus Index Bus Index M1 M4 M1 M4 M2 M5 M2 M5 M3 M3 (a) (b) Figure 4: EPRA result comparison on the IEEE 118-bus system. (a) Error of the mean. (b) Error of the standard deviation. (4) For M1–M5, because of the superior quality of the Table 4 and Figure 4, respectively. *e test sample number is GP (e.g., its small sample requirement), the training set to be the same as for the IEEE 30-bus system. *e results show that the proposed method can guar- sample size of the proposed method is much smaller than those of M1–M3, which aligns well with the antee EPRA accuracy even in a large case while improving current industry practice. the computational efficiency. It also shows that the statistical moments (i.e., the mean and standard deviation) diverge far *e errors of the EPRA results are compared, as shown from the real value at Buses 72∼110, which demonstrates in Figure 3. For the proposed method, the EPRA results for that EPRA cannot be achieved by data-driven direct the mean and standard deviation are very accurate. How- learning. ever, for M1∼M4, the estimation results for the mean are relatively accurate, while the standard deviations are far 5. Conclusions and Future Work from the real value obtained by M0, so they cannot achieve anaccurateEPRA. Inparticular, asseen inFigure3,the error *is paper proposes a data-driven framework for EPRA. is abnormally large in some specific buses (e.g., Bus 8 and Specifically, a GP surrogate model is developed to identify Bus 24). *is is because the LMPs at these buses fluctuate the marginal generators and congested transmission lines of much more than other buses, making it challenging to be DC-OPF. *is paves the way for improving the efficiency of directly learned by the data-driven methods, resulting in EPRA. Based on the KKTcondition, an adaptability criterion false information being passed to ISOs and market partic- is proposed to identify samples with learning errors. *e ipants, leading to severe market risks. simulation results show that the proposed method increases EPRA accuracy. Comparisons with recent data-driven 4.2. Results on the IEEE 118-Bus System. *e IEEE 118-bus methods show that the proposed approach can greatly system is used to demonstrate the scalability of the proposed improve the computational efficiency of EPRA without method. *e learning error and EPRA results are given in compromising its accuracy. It is also shown that direct Error of Mean (MW/$) Error of Standard Deviation (MW/$) 10 International Transactions on Electrical Energy Systems with various technologies,” Protection and Control of Modern learning for LMP may not be tractable due to the problem Power Systems, vol. 6, no. 1, pp. 37–54, 2021. complexity. Future work will consider more uncertain [4] J. Huang, Y. Xue, Z. Y. Dong, and K. P. 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Published: Jan 31, 2022

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