17th Power Systems Computation Conference (PSCC'11), Stockholm, Sweden

A Piece-wise Linear Model For Approximating LMP-Load Curve Based On Critical Load Levels

Fangxing Lia, Rui Bob aUniversity of Tennessee, Knoxville,USA bMISO, St.Paul, USA August 23, 2011 1

Outline








Background Problem Statement Possible Approaches Critical Load Level (CLL) and its Identification Piece-wise Linear Approximation Model Numeric Study Conclusions 2

Background


Locational Marginal Price (LMP)


Load level is a major factor LMPi

D∑ LMPi = f(D∑)




Knowing function f is useful for analytical LMP Studies


Probabilistic LMP Forecasting





E.g.: impact of load forecasting error on LMP

LMP Trend Analysis 3

Background (Cont’d)


Need to know function f





LMPi = f(D ) ∑

However, …


Can be highly nonlinear





High loss cases

Multiple patterns


Step change; linear; polynomial; nonlinear

Difficult to define a mathematical representation




4

Problem Statement


Define a simple form of math representation for the LMP-Load curve with high accuracy

5

Possible Approaches


Data-driven approach






E.g.: Curve-fitting, Curve-smoothing Difficult to define break points of drastic pattern change

We propose model-driven approach


LMP change its pattern when









Shift of marginal units Change in congestions

These break points have been studied and termed Critical Load Level (CLL)! With CLLs as breakpoints, even linear model may work! 6

Critical Load Level (CLL) and its Identification


CLL is critical because








Across a CLL system statuses may change significantly from their past trajectory E.g.: shift of marginal unit; new congestion; etc.

Locating CLL


Criterion: A system load level is a Critical Load Level

where the binding constraint set or non-binding constraint set changes when system load varies right across this load level.


Methods




Binary Search Method Simplex-like Method (for no-loss case): Reference [5] Variable Substitution Method (for loss case): Reference [14]

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Piece-wise Linear Approximation Model


Steps






1) Identify CLLs using existing methods 2) Perform linear curve-fitting for any segment demarcated by two immediately adjacent CLLs 3) Combine the piece-wise linear models to get a full representation

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Piece-wise Linear Approximation Model (Cont’d)

D0 < D ≤ D1 a 0 × D + b0 , a × D + b , D1 < D ≤ D2 1  1 y ( D) =  M a × D + b , D < D ≤ D n −1 n −1 n  n −1 a n × D + bn , Dn < D ≤ Dn +1

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Numeric Study


IEEE 118-bus system



Studied load range: 42.42MW~7686MW Benchmark LMP is obtained from repetitive AC-OPF runs





Sampling step: 4.242MW (i.e., 0.001 p.u. of base case load)

Proposed Piece-wise linear model


57 CLLs identified through binary search method







To the precision of 4.242MW Can use more effective methods

Do linear curve-fitting for each of the 58 segments Resulting math formula is a piece-wise linear model with 57 break points (i.e., 58 segments) 10

Results

Benchmark LMP-Load Curve

Approximated LMP-Load Curve using proposed piece-wise linear model

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Close-up Look --- Step Change Pattern 42.42MW-1000MW

Benchmark LMP-Load Curve

Approximated LMP-Load Curve using proposed piece-wise linear model

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Close-up Look --- Polynomial Pattern 1000MW-2000MW

Benchmark LMP-Load Curve

Approximated LMP-Load Curve using proposed piece-wise linear model

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Close-up Look --- Linear Pattern 5000MW-6000MW

Benchmark LMP-Load Curve

Approximated LMP-Load Curve using proposed piece-wise linear model

14

Close-up Look --- Complex Pattern 7000MW-7700MW

Benchmark LMP-Load Curve

Approximated LMP-Load Curve using proposed piece-wise linear model

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Conclusions


Proposed Piece-wise Linear Model







Sufficiently accurate approximation to actual LMPLoad curve Able to capture various pattern of the curve Key to success: identification and utilization of CLLs LMP-Load curve can take math form as simple as piece-wise linear model


Greatly facilitate sophisticated analytical studies

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Q&A

Questions?

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