A Maximum A-Posteriori Based Algorithm for Dynamic Load Model - - PowerPoint PPT Presentation

A Maximum A-Posteriori Based Algorithm for Dynamic Load Model Parameter Estimation

Siming Guo and Prof. Thomas Overbye

sguo6@illinois.edu September 21, 2015

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Source: http://www.tva.gov/power/rightofway/images/high_cost_tree.jpg, http://home.iitk.ac.in/~ankushar/rtds/images/sel451.jpg, http://www.ee.washington.edu/research/pstca/pf30/pg_tca30fig.htm

0.2 0.4 0.6 0.8 0.85 0.9 0.95 1 1.05 Time [s] Voltage [pu] 0.2 0.4 0.6 0.8 0.85 0.9 0.95 1 1.05 Time [s] Voltage [pu]

PowerWorld Load model Compare

Measurement based parameter estimation

argmin

𝑞

𝑤𝑛𝑓𝑏𝑡 − 𝑤𝑞 2

2 Simulation process Simulation vp Load model p Ideal Simulation process Simulation vp Load model p Simulation non-injective Simulation process Simulation vp Load model p

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Test case

Source: PSS/E 33.5 Model Library

𝑞 = %𝑀𝑁 %𝑇𝑁 %𝐸𝑀 %𝐷𝑄 %𝑄𝐽/𝑅𝑎

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Impact of measurement noise

𝑤𝑛𝑓𝑏𝑡 − 𝑤𝑞

2 2 is insensitive to parameters

Parameter estimate is very sensitive to noise

George E. P. Box: “Essentially, all models are wrong, but some are useful”

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Prediction accuracy

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Solution 1: Use multiple disturbances

argmin

𝑞

𝑤𝑛𝑓𝑏𝑡,𝑔𝑏𝑣𝑚𝑢 1 − 𝑤𝑞,𝑔𝑏𝑣𝑚𝑢 1 2

2 + 𝑤𝑛𝑓𝑏𝑡,𝑔𝑏𝑣𝑚𝑢 2 − 𝑤𝑞,𝑔𝑏𝑣𝑚𝑢 2 2 2

Simulation 1 vp Load model p Simulation process Simulation 2 vp Simulation process

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Solution 1: Results

8 / 11 𝑔

𝑊

𝑔

𝑄

argmax

𝑞

Pr{𝑞|𝑤𝑛𝑓𝑏𝑡} = argmax

𝑞

Pr 𝑤𝑛𝑓𝑏𝑡 𝑞 ∙ Pr{𝑞} Pr{𝑤𝑛𝑓𝑏𝑡} = argmax

𝑞

Pr 𝑤𝑛𝑓𝑏𝑡 𝑞 ∙ Pr{𝑞}

𝑢=1 𝑈

𝑔

𝑊(𝑤𝑛𝑓𝑏𝑡 𝑢 − 𝑤𝑞 𝑢 ) 𝑜=1 𝑂

𝑔

𝑄(𝑞 𝑜 − 𝜈𝑞 𝑜 )

Solution 2: Maximum a-posteriori (MAP) estimator

Simulation vp Load model p Simulation process Simulation vp Load model p Simulation process

𝜈𝑞 𝑜 𝑞 𝑜 𝑤𝑛𝑓𝑏𝑡 𝑢 𝑤𝑞 𝑢

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Solution 2: Implementation issue

≅

𝑢=1 𝑈

𝑔

𝑊(𝑤𝑛𝑓𝑏𝑡 𝑢 − 𝑤𝑞 𝑢 ) 1 𝑈

=

𝑢=1 𝑈

1 2𝑐 exp − 𝑤𝑛𝑓𝑏𝑡 𝑢 − 𝑤𝑞 𝑢 𝑐

1 𝑈

= 1 2𝑐 exp

𝑢=1 𝑈

− 𝑤𝑛𝑓𝑏𝑡 𝑢 − 𝑤𝑞 𝑢 𝑐

1 𝑈

= 1 2𝑐 exp 1 𝑈

𝑢=1 𝑈

− 𝑤𝑛𝑓𝑏𝑡 𝑢 − 𝑤𝑞 𝑢 𝑐 Because: argmax

𝑞

Pr 𝑤𝑛𝑓𝑏𝑡 𝑞 ∙ Pr{𝑞} One problem:

• 𝑔

𝑊 1𝜏 = 0.17

• 30 seconds @ 30 samples/s

 𝑈 = 900

• 0.17900~10−693
• Smallest double precision

number ~10−308 ~1

𝑢=1 𝑈

𝑔

𝑊(𝑤𝑛𝑓𝑏𝑡 𝑢 − 𝑤𝑞 𝑢 )

𝑔

𝑊

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Solution 2: Results

Prior 𝑔

𝑄 dominates

Data 𝑔

𝑊 dominates

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Summary

Problem: Lack of injectivity leads to bad objective function… Solution: MAP estimator Injectivity in load modeling …which leads to poor predictions Implementation issues 0.17900~10−693