0

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754

2465

2210

0

0

0

25

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348

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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2465

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25

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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2465

2210

0

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25

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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Direct Non-Iterative Power System State Solution and Estimation

B. Fardanesh

NYPA

Advanced Energy Conference

New York, 2013

0

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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Basics of the Re-Linearization Method

X1

X2

f(x1,x2)

(X1*, X2*)

X1 X2

X12

X22

X1

X2

Ys and Zs

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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25

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348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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25

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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A Simple Example

• Redundant Polynomial Equations

• Dual Transformation and Direct One-Shot Solution

13125

11527

2635

1022

82

13222

21

32332

21

2232131

12232

22321

21

xxxxx

xxxxx

xxxxxx

xxxx

xxxxx

13

11

26

10

8

00121

5000

3501

0021

0000

50001

20107

00010

20000

01121

4

3

2

1

23

20

33

12

11

z

z

z

z

y

y

y

y

y

Basics of the Re-Linearization Method

If only one Z If Z1 and Z2 , we will have three ts

t

Y1(t)

Y2(t)

Y3(t)

t*

Y3(t1,t2 ,t3)=0

t1

t2(t1*, t2*, t3*)

t3

Y2(t1,t2 ,t3)=0

Y1(t1,t2 ,t3)=0

0

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25

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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Basics of the Re-Linearization Method

jminjnimmnij yyyyyy ....Form the hyperplanes:

24

23

22

214342324131214321 zzzzzzzzzzzzzzzzzzzz

In term of Zs and change variables to ts:

Vi2

Vi

Vj2

VjVi Vj

Vk2

Vk

Vj2

Vj

Vi2

Vi

Vi Vj

Vj Vk

Vi Vk

Vk Vl

Vi Vk

Vi Vj

Vj2

Vj

Vi2

Vi

Vk2

Vk

Vj Vl

Vl2

Vl

Relation to Network Topology

0

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25

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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348

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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25

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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2465

2210

0

0

0

25

60

348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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• Historically we have used iterative techniques to solve the state estimation problem

• A direct one-shot solution for the state of a power system is now possible

• Full AC solution-No Simplifications• No more iterations• No reliance on the “goodness” of the initial

guess• An envisioned faster more robust solution

New Paradigm

0

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2210

0

0

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25

60

348

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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0

145

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754

2465

2210

0

0

0

25

60

348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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145

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2465

2210

0

0

0

25

60

348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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145

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2465

2210

0

0

0

25

60

348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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What is Required?

• Accurate measurements from the system at the control center

• Bus voltage phasor and line current phasor measurements

• Locally (at the substation) validated data– “Super-calibrator “ a plus

• Reliable and redundant communications network

0

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2210

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1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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0

145

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754

2465

2210

0

0

0

25

60

348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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0

145

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754

2465

2210

0

0

0

25

60

348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

13

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5

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t

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0

145

130

754

2465

2210

0

0

0

25

60

348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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A Power System State Estimator/Solver

• Power Flow equations in Rectangular from• Naturally in the desired form:

• Measurement equations have similar form

NjQQbbaaYbaaaY

PPbabaYbbaaY

jj

jj

DGijjiijjiij

N

liij

DGijjiijijij

N

liij

,2.Im.Re

.Im.Re

iV =ai + jbi

0

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2465

2210

0

0

0

25

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348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

13

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3

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1

t

t

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0

145

130

754

2465

2210

0

0

0

25

60

348

676

1020

000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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000001715.10012000

00100501105.05605505295.64

0005.00355.15.111525268

350010353005003160101611152600231

5011700955038211704095212055850118935.1125

15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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Envisioned Benefits• Direct State Estimation• More robust– No more iterations• No reliance on the initial guess• Fast– Perhaps limited only by the

communication links’ latency• A much more “dynamic” assessment of the

system conditions and behavior• Potential for ultimate use in closed-loop and

automatic control of power systems

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000001715.10012000

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0005.00355.15.111525268

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15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

14

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000001715.10012000

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0005.00355.15.111525268

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15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

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000001715.10012000

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0005.00355.15.111525268

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15005.9253515865.23481173854303042162

3010500105.026050

31000401001100262900

001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

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15005.9253515865.23481173854303042162

3010500105.026050

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001170150005.900850

00175.900050118001090

051775.03000110026975.13

001700150201755.1153211212016675.163

92505.0300061017156260064

01198925.1408505.7033760028495.241

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NYPA ECC Implementation• Snapshot from the Siemens EMS• Reduced NYS data—230kV and above• Solve for bus voltage phasors one bus away

from each PMU• Purely phasor based direct SE• Output can be utilized as “anchor points: for

the traditional SE• Will report on the performance soon