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

Measurement based parameter estimation 1.05 1 Voltage [pu] 0.95 0.9 0.85 0 0.2 0.4 0.6 0.8 Time [s] 2 argmin 𝑤 𝑛𝑓𝑏𝑡 − 𝑤 𝑞 2 Compare 𝑞 Load model Load model Load model Simulation Simulation Simulation p p p v p v p v p 1.05 non-injective 1 Simulation Voltage [pu] Ideal 0.95 Load model 0.9 PowerWorld 0.85 0 0.2 0.4 0.6 0.8 Time [s] Simulation Simulation Simulation process process process 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 2 / 11

Test case 𝑞 = %𝑀𝑁 %𝑇𝑁 %𝐸𝑀 %𝐷𝑄 %𝑄𝐽/𝑅𝑎 Source: PSS/E 33.5 Model Library 3 / 11

Impact of measurement noise 2 is insensitive to parameters 𝑤 𝑛𝑓𝑏𝑡 − 𝑤 𝑞 Parameter estimate is very sensitive to noise 2 George E. P. Box: “Essentially , all models are wrong, but some are useful” 4 / 11

Prediction accuracy 5 / 11

Solution 1: Use multiple disturbances Simulation 1 Load model Simulation 2 v p v p p Simulation Simulation process process 2 + 𝑤 𝑛𝑓𝑏𝑡,𝑔𝑏𝑣𝑚𝑢 2 − 𝑤 𝑞,𝑔𝑏𝑣𝑚𝑢 2 2 2 argmin 𝑤 𝑛𝑓𝑏𝑡,𝑔𝑏𝑣𝑚𝑢 1 − 𝑤 𝑞,𝑔𝑏𝑣𝑚𝑢 1 2 𝑞 6 / 11

Solution 1: Results 7 / 11

Solution 2: Maximum a-posteriori (MAP) estimator Simulation Simulation Load model Load model argmax Pr{𝑞|𝑤 𝑛𝑓𝑏𝑡 } v p v p p p 𝑞 Pr 𝑤 𝑛𝑓𝑏𝑡 𝑞 ∙ Pr{𝑞} = argmax Pr{𝑤 𝑛𝑓𝑏𝑡 } 𝑞 = argmax Pr 𝑤 𝑛𝑓𝑏𝑡 𝑞 ∙ Pr{𝑞} Simulation Simulation 𝑞 process process 𝑈 𝑂 𝑔 𝑊 (𝑤 𝑛𝑓𝑏𝑡 𝑢 − 𝑤 𝑞 𝑢 ) 𝑔 𝑄 (𝑞 𝑜 − 𝜈 𝑞 𝑜 ) 𝑢=1 𝑜=1 𝑔 𝑔 𝑊 𝑄 𝑤 𝑛𝑓𝑏𝑡 𝑢 𝑤 𝑞 𝑢 𝜈 𝑞 𝑜 𝑞 𝑜 8 / 11

Solution 2: Implementation issue 1 𝑈 𝑈 ≅ 𝑔 𝑊 (𝑤 𝑛𝑓𝑏𝑡 𝑢 − 𝑤 𝑞 𝑢 ) 𝑢=1 1 𝑈 𝑈 2𝑐 exp − 𝑤 𝑛𝑓𝑏𝑡 𝑢 − 𝑤 𝑞 𝑢 1 One problem: Because: argmax Pr 𝑤 𝑛𝑓𝑏𝑡 𝑞 ∙ Pr{𝑞} = 𝑐 𝑞 • 𝑔 𝑊 1𝜏 = 0.17 𝑢=1 1 • 30 seconds @ 30 samples/s 𝑈 𝑈 = 1 − 𝑤 𝑛𝑓𝑏𝑡 𝑢 − 𝑤 𝑞 𝑢  𝑈 = 900 𝑈 2𝑐 exp 𝑐 0.17 900 ~10 −693 𝑔 𝑊 (𝑤 𝑛𝑓𝑏𝑡 𝑢 − 𝑤 𝑞 𝑢 ) • 𝑢=1 𝑈 𝑢=1 • Smallest double precision − 𝑤 𝑛𝑓𝑏𝑡 𝑢 − 𝑤 𝑞 𝑢 = 1 1 𝑔 2𝑐 exp 𝑈 number ~10 −308 𝑊 𝑐 𝑢=1 ~1 9 / 11

Solution 2: Results Prior 𝑔 Data 𝑔 𝑄 dominates 𝑊 dominates 10 / 11

Summary Injectivity in load modeling Problem: Lack of injectivity leads to bad objective function… …which leads to poor predictions Solution: MAP estimator 0.17 900 ~10 −693 Implementation issues 11 / 11