Redundancy analysis and improved parameter

estimation for complex dynamic load modeling:

A case study on the WECC CMPLDW

Kaiqing Zhang

Graduate Research Assistant

Power & Energy Systems Group

University of Illinois, Urbana-Champaign

Feb. 22, 2015

Motivation

▪ Load modeling traditionally less investigated compared

to G&T

▪ Siemens PSS/E’s dynamic model library has

– > 100 GEN models

– ~40 Transmission models

– Only 9 load models

▪ Several existing dynamic studies attribute the

uncertainty of their solutions to load models adopted

[De Leon et al’06]

▪ Increasing stresses on the grid will exacerbate this

situation

2

Measurement-based load modeling

▪ Wide deployment of digital fault recorders (DFRs)

enables more accurate load modeling via data fitting

[EPRI’06, LNBL ’10]

▪ Fault-induced delayed voltage recovery (FIDVR) typed

disturbance events of particular interest [NERC

report’09, Liu et al’13]

3

Load model

g(𝐕, 𝐟, 𝛉)

Output P &

Q

Mismatch 𝐫

Measurements 𝐲

Voltage

profile 𝐕Frequency 𝐟

▪ Nonlinear least-squares (NLS)

min𝛉

𝐶 𝛉 ≔1

2𝐫 𝛉 2

2 =1

2𝐲 − 𝐠(𝛉) 2

2,

▪ Approach: iterative Levenberg-Marquardt (L-M)∆𝛉 = − 𝐉T𝐉 + λ𝐃T𝐃

−1(𝐉T𝐫)

Composite load models

▪ Aggregated by several load components [Choi et al’09]

▪ Parameter identifiability [Ma et al’08, Kim et al’16]

▪ Numerical issues in data fitting

– Slow convergence

– Parameter evaporation

4

M. K. Transtrum, and J. P. Sethna, “Improvements to the Levenberg-Marquardt

algorithm for nonlinear least-squares minimization,” Preprint submitted to Journal of

Computational Physics, pp. 32, Jan. 2012

0 10 20 30 40 500

10

20

30

40

50

Large Induction Motor %

Co

nst

ant

Po

wer

%

Correctvalues

Figure from Siming

Our focus

▪ Develop a systematic framework to quantify/visualize

redundancy among multiple parameters for any

dynamic load model

– Existing approaches in [Son et al ’14], and [Kim et al ’15]

limited to the CIM6+ZIP model and can only identify

pairwise correlation

▪ Improve the numerical performance of iterative NLS

approaches to be more robust against potential

plateau

– By excluding trivial parameters

– By incorporating a prior knowledge

5

IEEE CIM6+ZIP

The WECC CMPLDW

▪ An improved load model to better capture load dynamics

during FIDVR

▪ Six load components (4 motors, electronic load, ZIP load)

▪ About 130 parameters in total

▪ Nominal parameter setups [Borden et al’10]6

CMPLDW Block Diagram [Source: Powerworld Block Diagram LoadCharacteristic]

Parameter Sensitivity Analysis ▪ Dynamic characteristics-related parameters

– Loads:

• Fractions: FmA, FmB, FmC, FmD, Fel

• 3-Φ motors: e.g., LFm, Ls, Lp, Lpp, Tpo, Tppo, H, Vtr1, Ttr1, Ftr1, Vrc1,

Trc1

• 1-Φ motors: e.g., CompPF, Vstall, Rstall, Xstall, Tstall, Vrst, Trst, Tth,

Th1t, Tvd

• Electronic load: e.g., PFel, Vd1, Vd2, frcel

• Static load: e.g., PFs, P1e, P1c, Pfrq, Q1e, Q1c, Qfrq

– Feeder: e.g., Rfdr, Xfdr, Fb

– Substations: Xxf

▪ Trajectory sensitivity [Hiskens-Pai’00] approximation

– Data source : PQube dataset provided by Southern California

Edison (SCE)

– Finite-difference derivative approximation

𝐽 𝑡 =𝜕𝑓(θ, 𝑡)

𝜕θ≈𝑓 θ + ε, 𝑡 − 𝑓 θ − ε, 𝑡

2ε7

Parameter Sensitivity Analysis (Cont.)

▪ Root-mean-squares (RMS) of trajectory sensitivities

▪ Parameter sensitivity depends on both the fault and

the initial parameter values

▪ About 60 sensitive parameters are included for further

analysis

8

RMS of sensitivities using SCE Data 5 RMS of sensitivities using SCE Data 6

Singular Value Decomposition

▪ Center and normalize the Jacobian J𝑠 of sensitive

parameters

▪ Singular value decomposition (SVD)

J𝑠 = U ∙ Σ ∙ VT

▪ J𝑠 is of low-rank, indicating parameter dependencies

▪ Truncated J𝑀𝑠 = U𝑀Σ𝑀V𝑀

T

▪ Define the scaled particip-

ation feature (SPF)

‒ F𝑖 sign flipping

F𝑖 = Σ𝑀V𝑀Te𝑖

9

principal

components

singular

values

participation

loadings

Dependency Detection and Visualization

▪ Clustering based parameter dependency detection

▪ K-Medoids [Kaufman ’09]

‒ Choose a data point instead of the average as the center

‒ Minimize the sum of pairwise dissimilarities

‒ More robust to noises and outliers

▪ Clustering evaluation index : e.g., Silhouette

[Rousseeuw ’87]

𝑠 𝑖 =𝑏 𝑖 − 𝑎(𝑖)

max{𝑎 𝑖 , 𝑏(𝑖)}

▪ Multidimensional scaling (MDS) [Borg-Groenen ’03]

‒ A lower-dimensional representation that preserves pair-

wise distances

‒ For dependency visualization

‒ For clustering results validation10

Choosing Optimal 𝑘▪ Silhouette across clusters when 𝑘 = 10, 𝑘 = 20, 𝑘 = 30

11

𝑘 = 10 𝑘 = 20 𝑘 = 30

Clustering Results Validation

12

▪ Trajectory sensitivities and clustering results visualized

by MDS when 𝑘 = 203Φ B&C 2nd low voltage trip delay time

3Φ B&C 2nd low voltage reconnection delay time

3Φ B&C 1st low voltage reconnection level

3Φ B&C synchronous reactance

1Φ D stall reactance

1Φ D thermal protection trip level

1Φ D thermal time constant

Improved Parameter Estimation

13

▪ Regularized NLS

– Slow convergence v.s. Parameter evaporation

– Objective regularization

min𝛉

𝐶 𝛉 =1

2‖𝐫 𝛉 ‖2

2 +1

2μ‖𝚪 ∙ (𝛉 − 𝛉𝟎)‖2

2,

– A prior knowledge of parameters 𝛉0 from previous tests

or load survey

– Regularization coefficient μ > 0 and weighting matrix 𝚪

– Levenberg-Marquardt step

∆𝛉 = − 𝐉T𝐉 + μ𝚪 + λ𝐃T𝐃−1

𝐉T𝐫 + μ𝚪 ∙ (𝛉 − 𝛉0)

Simulation on Synthetic Measurements

14

Output fitting results for the SCE Data 5, using synthetic P and Q data

with no noises

Comparison of Parameter Values

15

'Rstalld' 'Rfdr' 'Xfdr' 'FmD' 'Xxf' 'TTr1d' 'Vrc2a'

Synthetic 0.5 0.04 0.05 0.2 0.01 0.02 0.8

L-M 0.1916583 0.0503820

4

0.0569938

7

0.1359223 0.072381

0

0.0197794 0.6553204

Reg-L-M 0.4631009 0.0730058

4

0.0510472

9

0.1297006 0.020114

7

0.0207914 0.646751'Vrc1c' 'Ttr2c' 'Ttr1c' 'Lpa' 'FmB' 'Th1td' 'Ftr1b'

Synthetic 0.65 0.02 0.02 0.1 0.2 0.3 0.2

L-M 0.6875355 0.0196519

4

0.0195235

1

0.1874902

5

0.2563025 0.01 0.0407743

7

Reg-L-M 0.687564 0.0201532

4

0.0200005

1

0.1718149 0.18208 0.2847848 0.1117401

'Th2td' 'FmA' 'Vd1' 'Trc2a' 'Rsb' 'Tpoc' 'Tpoa'

Synthetic 1.7 0.05 0.75 0.73 0.03 0.1 0.092

L-M 1.092873 0.2185689 1.150139 1.11035 0.0158193

6

0.0667506

4

0.1974019

Reg-L-M 1.974133 0.0650879

6

0.8327271 0.976271 0.0493635

9

0.1306721 0.1001608'Trc1b' 'Hc' 'Lsc' 'P2e' 'Q2c' 'Pfrq 'Qfrq'

Synthetic 0.6 0.1 1.8 1 1.5 -1 -1

L-M 0.8753621 0.01 1.281682 4.144712 2.672409 -62.43499 -7.97859

Reg-L-M 0.6016281 0.0798483

1

1.845069 1.018427 1.502463 -0.9999593 -0.9999132

Simulation on Noisy Measurements

16

Output fitting results for the SCE Data 5, using synthetic P and Q data

with noises

Simulation on Real Measurements

17

Output fitting results for the SCE Data 6, using real P and Q

measurements data

Comparison of Parameter Values

18

'Ttr2c' 'Rfdr' 'Xfdr' 'Xxf' 'FmC' 'Vtr2c' 'FmA'

L-M 0.01974704 0.0384908

90.05377107 0.09259105 0.09772088 0.7266558 0.0876438

Reg-L-M 0.01973941 0.0483073 0.0522862

9

0.0689083

9

0.1114887 0.7237592 0.1163153

'FmD' 'Vtr2b' 'Vd2' 'PFs' 'Ftr2b' 'Vd1' 'CompPFd'

L-M 0.1981026 0.7381233 0.6281592 -0.9013258 0.2674962 0.7036738 1.0

Reg-L-M 0.2605251 0.6911023 0.6698054 -0.6526318 0.2970149 0.7066146 1.0

'Fb' 'LFmd' 'Lpa' 'Fel' 'Tpoc' 'PFel' 'Lpc'

L-M 0.7473184 1.0 0.1153315 0.2395047 0.09024712 1.0 0.1811523

Reg-L-M 0.7292499 1.0 0.04 0.0340158

4

0.1443441 1.0 0.4486523

'Rsb' 'Rsa' 'Hc' 'LFmc' 'P1c' 'P1e' ‘Q1c'

L-M 0.07069106 0.05025693 0.06846003 0.4044096 0.5738456 2.247874 0.7583346

Reg-L-M 0.4131136 0.2395168 0.01 0.4327505 0.6512989 2.071328 0.02354476

‘Q1e' 'Lsb' 'Tpob' 'Tpoa' 'Etrqb' 'Pfrq' 'Qfrq'

L-M 8.432362 1.918263 0.04610666 0.01302225-

4.223569-46.39091 85.85966

Reg-L-M 2.005963 1.771487 0.2009874 0.0091385 1.98918 -0.999848 -0.999829

Summary of Simulation Results

19

▪ A prior knowledge of parameters can help achieve

unique and reasonable parameter values

▪ Regularized L-M outperforms L-M in terms of

mismatches while effectively avoiding erroneous cases

when noisy measurements are applied

▪ The more sensitive a parameter is, the more

consistently it can be estimated

Conclusions

20

▪ There exists a high degree of redundancy in the WECC

CMPLDW parameters, challenging measurement-

based load modeling

▪ Quantifying and visualizing the redundancy suggests

it is difficult to estimate all parameters solely based on

measurement data

▪ Regularization can help achieve better parameter

identification performance by incorporating a prior

knowledge

▪ Future work

– Selection of μ and 𝚪

– Incorporating known feasibility constraints

– Real data based testing and validation

Acknowledgement

Professor Hao Zhu

Siming Guo

21

Questions

References

22

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