Learning the LMP-Load Coupling From Data Xinbo Geng Department of Electrical and Computer Engineering Texas A&M University [email protected] people.tamu.edu/∼gengxbtamu November 2, 2016 Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 1 / 23 

Introduction Highlights Background: DCOPF-based electricity market operations, Security-constrained Economic Dispatch. A Key Question: What is the relationship between nodal loads and Locational Marginal Prices (LMPs). Understand the Data: From a market participant’s viewpoint: understand the (load+price) data. Key References Geng, Xinbo, and Le Xie. “Learning the LMP-Load Coupling From Data: A Support Vector Machine Based Approach.” IEEE Transactions on Power Systems (accepted, to appear). Geng, Xinbo, and Le Xie. “A data-driven approach to identifying system pattern regions in market operations.” 2015 IEEE Power & Energy Society General Meeting. IEEE, 2015. Geng, Xinbo, and Le Xie. “Learning the LMP-Load Coupling From Data: A Support Vector Machine Based Approach.” arXiv preprint arXiv:1603.07276 (2016). (With complete technical details and many illustrative examples.) Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 2 / 23 

Problem Formulation Motivations of Understanding the LMP-Load Coupling Question: what is the relationship between loads and LMPs? Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 3 / 23 

Problem Formulation Motivations of Understanding the LMP-Load Coupling Question: what is the relationship between loads and LMPs? The relationship between Locational Marginal Prices (LMPs) and Loads A fundamental question. Understand electricity market operations. Benefit both system operators and market participants. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 3 / 23 

Problem Formulation Motivations of Understanding the LMP-Load Coupling Question: what is the relationship between loads and LMPs? The relationship between Locational Marginal Prices (LMPs) and Loads A fundamental question. Understand electricity market operations. Benefit both system operators and market participants. Becoming more and more important... Renewable Energy Source: blogs.scientificamerican.com Demand Response Source: nature.com Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 3 / 23 

Problem Formulation Intuition Monte-Carlo Simulation on a Simpler 3-bus System (1) Regard PD2 and PD3 as parameters, randomly generate PD2 and PD3 ; (2) solve the optimization problem (SCED); (3) record the Locational Marginal Prices. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 4 / 23 

Problem Formulation Intuition Monte-Carlo Simulation on a Simpler 3-bus System (1) Regard PD2 and PD3 as parameters, randomly generate PD2 and PD3 ; (2) solve the optimization problem (SCED); (3) record the Locational Marginal Prices. LMPs are discrete (3 possibilities). There are some disjoint regions. Load vectors of the same region leads to the same LMP vector. Xinbo Geng and Le Xie, “A Data-driven Approach to Identifying System Pattern Regions in Market Operations,” in IEEE Power and Energy Society General Meeting, 2015. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 4 / 23 

Theoretical Results Understand the LMP-Load Coupling 1 Introduction 2 Problem Formulation 3 Theoretical Results 4 Understand the Data 5 Conclusions Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 5 / 23 

Theoretical Results Key Definitions Everything here is based on Multi-parametric Linear Programming (MLP) Theory Definition (Optimal Partition/System Pattern) The system pattern π is defined as π := (B, N). B: indices of binding constraints; N: indices of non-binding constraints. B ∩ N = ∅, B ∪ N = J . J = {1, 2, · · · , nc }: the index set of constraints. system pattern: π = (B, N) represents the status of the system. system pattern = optimal partition (in MLP theory). Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 6 / 23 

Theoretical Results Key Definitions Everything here is based on Multi-parametric Linear Programming (MLP) Theory Definition (Optimal Partition/System Pattern) The system pattern π is defined as π := (B, N). B: indices of binding constraints; N: indices of non-binding constraints. B ∩ N = ∅, B ∪ N = J . J = {1, 2, · · · , nc }: the index set of constraints. system pattern: π = (B, N) represents the status of the system. system pattern = optimal partition (in MLP theory). Definition (Critical Region/System Pattern Region) System Pattern Region (SPR) refers to the set of load vectors which lead to the same system pattern π = (Bπ , Nπ ): Sπ := {PD ∈ D : B(PD ) = Bπ } (1) system pattern region = critical region (in MLP theory). Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 6 / 23 

Theoretical Results Load Space Decomposition Characteristics of SPRs Theorem (Load Space Decomposition) The load space could be decomposed into many SPRs. Each SPR is a convex polytope. The relative interiors of SPRs are disjoint convex sets and each corresponds to a unique system pattern a. a Zhou, Q., Tesfatsion, L., & Liu, C. Short-term congestion forecasting in wholesale power markets. IEEE Trans on Power Systems Analyze the 3-bus system using the Multi-parametric Programming Toolbox 3.0: 10 SPRs. Every SPR is convex. SPRs are disjoint E.g. SPR#5: No congestion. Marginal generator: gen#1. E.g. SPR#3: Line 1 → 3 is congested. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 7 / 23 

Theoretical Results LMP-Load Coupling Locational Marginal Prices (LMPs) and System Pattern Regions (SPRs) Theorem (LMPs Remain the Same in an SPR) Within each SPR, the vector of LMPs is uniquea. a Xinbo Geng, & Le Xie (2015). A Data-driven Approach to Identifying System Pattern Regions in Market Operations. In IEEE Power and Energy Society General Meeting. E.g. SPR#5: LMP: λ = [20, 20, 20]. E.g. SPR#3: LMP: λ = [20, 50, 80]. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 8 / 23 

Theoretical Results LMP-Load Coupling Locational Marginal Prices (LMPs) and System Pattern Regions (SPRs) Theorem (Distinct LMP Vectors) Different SPRs have different LMP vectors. a a Xinbo Geng, Le Xie. “Learning the LMP-Load Coupling From Data: A Support Vector Machine Based Approach” IEEE Transactions on Power Systems (Accepted) Each SPR has a unique LMP vector. E.g. SPR#5: LMP: λ = [20, 20, 20]. E.g. SPR#3: LMP: λ = [20, 50, 80]. The boundary between two SPRs is linear. Lemma (Separating Hyperplanes) Because SPRs are convex sets, there exists a separating hyperplane between any two SPRs. Xinbo Geng, & Le Xie (2015). A Data-driven Approach to Identifying System Pattern Regions in Market Operations. In IEEE Power and Energy Society General Meeting. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 9 / 23 

Understand the Data A Data-driven Approach For Market Participants 1 Introduction 2 Problem Formulation 3 Theoretical Results 4 Understand the Data 5 Conclusions Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 10 / 23 

Understand the Data From the Viewpoint of System Operators Understand the LMP-Load Coupling Understanding the System Pattern Regions ⇔ understanding the LMP-Load Coupling. For the System Operators They know EVERYTHING (Confidential Information:) System Topology. Transmission Limits. Line Parameters. Generation costs. ... They can analytically calculate the System Pattern Regions (SPRs). (e.g. Multi-parametric Programming Toolbox.) Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 11 / 23 

Understand the Data From A Market Participant’s Viewpoint How can the Market Participants Estimate the System Pattern Regions? They know almost NOTHING about Confidential Information (e.g. System Topology, Transmission Limits, Line Parameters, Generation costs.....) Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 12 / 23 

Understand the Data From A Market Participant’s Viewpoint How can the Market Participants Estimate the System Pattern Regions? They know almost NOTHING about Confidential Information (e.g. System Topology, Transmission Limits, Line Parameters, Generation costs.....) Answer: Data! Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 12 / 23 

Understand the Data From A Market Participant’s Viewpoint How can the Market Participants Estimate the System Pattern Regions? They know almost NOTHING about Confidential Information (e.g. System Topology, Transmission Limits, Line Parameters, Generation costs.....) Answer: Data! Figure : Load Data Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 12 / 23 

Understand the Data From A Market Participant’s Viewpoint How can the Market Participants Estimate the System Pattern Regions? They know almost NOTHING about Confidential Information (e.g. System Topology, Transmission Limits, Line Parameters, Generation costs.....) Answer: Data! Figure : Load Data Figure : Load Data and LMP Data Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 12 / 23 

Understand the Data SPR Identification Problem is a Classification Problem Model the SPR Identification Problem as a Classification Problem We proved: Load vectors within an SPR have the same LMP vectors. Different SPRs have different LMP vectors. Load Data. LMP Data. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 13 / 23 

Understand the Data SPR Identification Problem is a Classification Problem Model the SPR Identification Problem as a Classification Problem We proved: Load vectors within an SPR have the same LMP vectors. Different SPRs have different LMP vectors. Load Data. LMP Data. Classification Problem: Given a feature vector x, label x with a label vector y. Feature Vector: load vector. Label Vector: LMP vector. Identify SPRs ⇔ Identify the separating hyperplanes among SPRs Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 13 / 23 

Understand the Data SPR Identification with SVMs Binary, Separable Case Not consider varying parameters (e.g. line limits): separable case. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 14 / 23 

Understand the Data SPR Identification with SVMs Binary, Separable Case Not consider varying parameters (e.g. line limits): separable case. Binary: only two classes y ∈ {1, −1}. Separable: Eqn. (3) is feasible. Separating Hyperplane: w PD − b = 0 . width of the margin: 2/||w||. Support Vector Machine (Binary, Separable Case) min w,b 1 2 w w (2) s.t y(i)(w P(i) D − b) ≥ 1, y(i) ∈ {−1, 1} (3) Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 14 / 23 

Understand the Data Multi-class SVM Classifier Two kinds multi-class SVM classifiers (n classes): “one-vs-all” pick one class, the rest n − 1 classes becomes another class, train a binary SVM classifier. Get n binary SVM classifiers. “one-vs-one” choose two classes out of n, train a binary classifier. Get n(n − 1)/2 binary SVM classifier. Existence of Separating Hyperplanes between any two SPRs ⇒ “one-vs-one” Max-vote-wins Given a load vector PD , each binary SVM classifier gives a vote (the index of SPR that PD belongs to), the SPR which gets the most vote is the final result. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 15 / 23 

Understand the Data SPRs with Varying System Parameters... Varying System Parameters Transmission line limits Generation limits Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 16 / 23 

Understand the Data SPRs with Varying System Parameters... Varying System Parameters Transmission line limits Generation limits (a) (54, 54, 72) (b) (60, 60, 80) (c) (66, 66, 68) Figure : System Pattern Regions With Different Transmission Limits Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 16 / 23 

Understand the Data Load and Price Data with Varying System Parameters Monte-Carlo Simulation, Again Transmission Line Limits are varying with time. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 17 / 23 

Understand the Data When System Pattern Regions Are Overlapping Binary, Non-separable Case s(i) slack variables (soft margins, tolerant errors). C i s(i): penalties of violation. Support Vector Machine (Binary, Non-separable Case) min w,b,s 1 2 w w + C i s(i) (4) s.t y(i)(w P(i) D − b) ≥ 1 − s(i), s(i) ≥ 0, y(i) ∈ {−1, 1} (5) Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 18 / 23 

Understand the Data Understand the Data in A Probabilistic Manner Posterior Probabilities (3-bus System) Estimating posterior probabilities with SVM: Binary: Platt Scaling. Multi-class Hastie & Tibshirani’s Algorithm. P(y = i|PD and y ∈ {1, 2, · · · , 8}), 8 classes, 8 surfaces. References: 1 Platt, John. “Probabilistic outputs for support vector machines and comparisons to regularized likelihood methods.” Advances in Large Margin Classifiers 10.3 (1999): 61-74. 2 Hastie, Trevor, and Robert Tibshirani. “Classification by pairwise coupling.” The Annals of Statistics 26.2 (1998): 451-471. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 19 / 23 

Conclusions Conclusions Summary Sketch the LMP-Load Coupling with Multi-parametric Programming Theory. Learn the LMP-Load Coupling From Data via Support Vector Machine. Model the SPR Identification as a Classification Problem. Understand the Data in a Probabilistic Manner (Fitting Posterior Probabilities). Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 20 / 23 

Conclusions Conclusions Summary Sketch the LMP-Load Coupling with Multi-parametric Programming Theory. Learn the LMP-Load Coupling From Data via Support Vector Machine. Model the SPR Identification as a Classification Problem. Understand the Data in a Probabilistic Manner (Fitting Posterior Probabilities). More Interesting Questions Impacts of Generation Costs. Partial Load Information. Discussed in the journal paper. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 20 / 23 

Conclusions Conclusions Summary Sketch the LMP-Load Coupling with Multi-parametric Programming Theory. Learn the LMP-Load Coupling From Data via Support Vector Machine. Model the SPR Identification as a Classification Problem. Understand the Data in a Probabilistic Manner (Fitting Posterior Probabilities). More Interesting Questions Impacts of Generation Costs. Partial Load Information. Discussed in the journal paper. Open Questions Impacts of Unit Commitment Decisions. Impacts of Ramp Constraints (multiple snapshots). Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 20 / 23 

Conclusions Any Questions? Learning the LMP-Load Coupling From Data Xinbo Geng Department of Electrical and Computer Engineering Texas A&M University [email protected] people.tamu.edu/∼gengxbtamu November 2, 2016 Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 21 / 23 

Conclusions Security-constrained Economic Dispatch (SCED) Objective: minimize total generation cost. Constraints: supply-demand balance, transmission limits, generation capacity limits. min PG nb i=1 ci (PGi ) s.t. nb i=1 PGi = nb j=1 PDj : λ1 F− ≤ H(PG − PD ) ≤ F+ : µ+, µ− P− G ≤ PG ≤ P+ G : η+, η− Assumptions: Lossless DC Optimal Power Flow (current practice). Quadratic Cost → Piecewise Linear (current practice). Assumptions (for simplicity) Piecewise Linear → Linear (WLOG). Each bus has (exactly) one load and one generator (WLOG). nb : number of buses. PG ∈ Rnb : generation vector. P+ G , P− G : generation limits. c ∈ Rnb : cost of generators. PD ∈ Rnb : load vector. F+, F−: transmission limits. Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 22 / 23 

Conclusions An Illustrative Example (3 Bus System) This 3-bus system serves as an illustrative example through this presentation. 3 Bus. 3 Transmission Lines. 3 Generators with linear cost. 2 Loads PD2 , PD3 . min PG1 ,PG2 ,PG3 20PG1 + 50PG2 + 100PG3 (6) s.t. PG1 + PG2 + PG3 = (PD1 ) + PD2 + PD3 (7) −   60 80 60   ≤   0 −2/3 −1/3 0 1/3 −1/3 0 −1/3 −2/3     PG1 (−PD1 ) PG2 − PD2 PG3 − PD3   ≤   60 80 60   (8) 0 ≤ PG1 ≤ 100, 0 ≤ PG2 ≤ 150, 0 ≤ PG3 ≤ 50 (9) Xinbo Geng (TAMU) LMP-Load-MLP-SVM November 2, 2016 23 / 23