Parallel kalman filter based time domain harmonic state estimation 160406

1. Rafael Cisneros-Magaña
Aurelio Medina-Rios
“Parallel Kalman Filter Based Time Domain
Harmonic State Estimation”
WORKSHOP
April 6, 2016
Morelia, México
Project C0014-2014-03 247099 Institutional Links CONACYT-British Council
“Modeling, analysis and digital/physical simulation of power systems with
integration of renewable energy sources; assessment of their dynamic
behavior and power quality impact”
Meeting, April 6, 2016
Morelia, México

2. 1. Introduction
The state estimation is now solved in time domain
using the parallel Kalman filter (PKF). This filter is
implemented using the CUDA platform and the
CUBLAS library on a GPU and is applied to estimate
harmonics and inter-harmonics.
The sequential parts of an algorithm are executed on
the CPU while the parts that are convenient to be
executed in parallel are run on the GPU using parallel
functions.
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3. 1. Int.
3
State Space Model: dx/dt=Ax+Bu
y=Cx+Du
Measurement Model: z=Hx
Discrete time:
x(k+1)=F(k)x(k)+B(k)u(k)
z(k)=H(k)x(k)

4. 2 Methodology
4
The PKF evaluates the time domain state
estimation using the following steps:
1. Data allocation in GPU memory.
2. Recursive execution of PKF on GPU.
3. The HSE result is saved.
4. The GPU memory is free when the case
study ends.

5. 2.1 Kalman Filter using CUBLAS library
5
Initial State x0, P0
Project Error Covariance Matrix 2.- P*(k)=F(k)P(k-1)F’(k)+Q(k)
KF Gain 3.- K=P*(k)H’(k)[H(k)P*(k)H’(k)+R(k)]-1
Update State 4.- x(k)=x*(k)+K[z(k)-H(k)x*(k)]
Update Error Covariance Matrix 5.- P(k) =[I-KH(k)]P*(k)
Project State 1.- x*(k)=F(k)x(k-1)+B(k)U(k)
Allocate GPU memory and set data from CPU to GPU

6. 2.1
6
KF Step CUBLAS functions
1 Dgemv, Daxpy
2 Dgemm, Dgeam
3 Dgemm, Dgeam, Dger, Dscal, Dtrsm
4 Dgemv, Daxpy
5 Dgemm, Dgeam
The CUBLAS functions to evaluate the KF steps are
shown. These functions are implemented in the
CUBLAS library to be executed in parallel form on the
GPU.

7. 3 Case Studies of Harmonic State Estimation
The IEEE 14 bus test system is modified to apply the PKF-
HSE method, state space and measurement models are
defined.
The test system is modified injecting harmonics and
interharmonics at buses 5 and 13. Bus voltages, line and load
currents are taken as state variables.
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8. 3.1 Case Study HSE with harmonic sources buses 5 and 13
Buses 5 and 13 receive injections of harmonics (5, 7, 11, 13).
The simulation time is 0.2 s. Actual, Kalman filter estimate and
difference for line currents are shown. The generators currents
are estimated.
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-0.2
0
0.2
0.4
0.6
Generator node 1
Current(pu)
Actual PKF Estimate PSCAD/EMTDC
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
0
1
2
Generator node 2
Time ( s )
Current(pu)
Actual PKF Estimate PSCAD/EMTDC

9. 3.1 Case Study
Table shows the harmonics injected at buses 5 and 13, the
harmonic spectrum of generator currents is calculated using
the DFT, the harmonic spectra agree with the injected
harmonics.
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Bus 5 13
Harmonic 5 7 11 13 5 7 11 13
Peak Value
Amp
6 3 1.5 0.75 3 1.5 0.75 0.37
3 5 7 9 11 13 15
0
1
2
3
4
5
6
7
Harmonic Order
%Fundamental
Generator Node 1
Generator Node 2

10. 3.2 HSE of time-varying harmonics
The PKF is applied under a varying harmonics condition,
Figures show the waveforms (actual, PKF estimate and
PSCAD) and the harmonic spectrum for generator current at
buses 1 and 2.
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-0.2
0
0.2
0.4
0.6
0.8
Generator node 1
Current(pu)
True PKF Estimate PSCAD/EMTDC
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
0
1
2
Generator node 2
Time ( s )
Current(pu)
True PKF Estimate PSCAD/EMTDC

11. 3.3 Case Study HSE including interharmonics.
The injected harmonics include inter-harmonics to verify the
effect on the system using the PKF-HSE. Figures show actual,
PKF estimate and PSCAD waveforms and the harmonic
spectrum of estimated generator currents.
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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-0.2
0
0.2
0.4
Generator node 1
Current(pu)
True PKF Estimate PSCAD/EMTDC
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-1
-0.5
0
0.5
1
1.5
2
Generator node 2
Time ( s )
Current(pu)
True PKF Estimate PSCAD/EMTDC
0 1 3 5 7 9 11 13 15
0
5
10
15
20
25
30
35
40
Harmonic Order
%Fundamental
Generator Node 1
Generator Node 2

12. 4. CPU-GPU execution time
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EXECUTION TIME (S)
Models CPU C++ code CPU-GPU CUBLAS Speed-up
1 36.2 22.5 1.6
2 434.7 52.2 8.3
3 1965 79.1 24.8
The power system model was replicated three times to
simulate larger systems, table presents the execution
time. The speed-up increases with the number of
models.

13. 5. Conclusions
• The HSE-PKF has been evaluated for
computational efficiency on the GPU to show
speed up against the sequential execution.
• Results are successfully compared against the
actual and PSCAD responses.
• A time domain harmonic state estimator based on
PKF using CUDA and CUBLAS on a GPU has been
presented.
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14. 14
Thank you!

15. 3.4 CPU-GPU Configuration and execution time
CPU: Intel Core TM i7-3770 CPU, 3.4 GHz, 16.0 GB RAM
GPU: NVIDIA GeForce GTX 680
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NVIDIA GEFORCE GTX 680 GPU DATA
Processor Cores 1536
Clock rate 1.08 Ghz
Memory 2 GB
Memory Clock rate 3 Ghz
Memory Bus Width 256 bit

16. 5 References
[1] J. D. Owens, M. Houston, D. Luebke, S. Green and J. E. Stone,
“GPU Computing”, Proc. IEEE, vol. 96, No. 5, pp. 879-899, May
2008.
[2] R. C. Dugan, M. F. McGranaghan, Surya Santoso and H. Wayne Beaty,
“Electrical Power Quality”, 2nd Ed., McGraw-Hill, 2002.
[3] IEEE Task Force on Harmonics Modeling and Simulation,
“Interharmonics: Theory and Modeling”, IEEE Trans. Power Del., vol 22,
No. 4, pp. 2335-2348, Oct., 2007.
[4] J. Arrillaga, N.R. Watson and S. Chen, Power System Quality
Assessment, John Wiley & Sons, 2000.
[5] K. Kennedy, G. Lightbody and R. Yacamini, “Power system harmonic
analysis using the Kalman filter”, IEEE Power Eng. Soc. General Meet.,
vol. 2, pp. 752-757, 2003.
[6] NVIDIA, CUDA C Programming Guide, Version 5.0, October 2012.
NVIDIA, CUDA API Reference Manual, Version 5.0, October 2012.
NVIDIA, CUDA Toolkit 5.0 CUBLAS Library, April 2012.
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17. 2.1 Parallel Kalman Filter algorithm using CUBLAS.
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The CUBLAS functions to implement LU are Dscal and
Dger. These functions evaluate the Crout’s reduction
algorithm and the Dtrsm function implements the
forward and backward substitutions. This step
consumes most of the time execution of the Kalman
filter and is calculated each time-step of the state
estimation.