**STATIC STATE ESTIMATION ALGORITHMS ** **USING HESSIAN MATRIX APPROACH**

**by**

**D U R G SIN G H C H A U H A N**

**T h e s is S u b m itte d in P a rtia l Fulfilm ent o f the -**
**R equirem ents fo r th e A w a rd o f the Degree o f **

**DOCTOR OF PH ILO SO PH Y ** **IN**

**E LECTR IC A L EN GIN EERIN G**

### ®* lh ^{»}

**Department of Electrical Engineering**

**IND IAN IN STITU TE OF TECH N O LO G Y, DELHI**

1985
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### TO

### LORD A Y Y A P P A

CERTIFICATE

This i s to c e r t if y that the th esis e n title d , " S ta tic State Estimation Algorithms Using Hessian Matrix .Approach", which i s submitted by Mr. Durg Singh Chauhan fo r the award o f the Degree o f Doctor o f Philosophy o f the Indian In s titu te o f Technology, New Delhi, i s a bonafide record o f research work carried out by him over the la s t three years under our guidance and supervision.

The candidate has f u l f i l l e d the requirements o f a ll the regulations rela tin g to the degree. The resu lts obtained in the th esis have not been submitted to any other U niversity or In stitu te fo r the award o f a degree or diploma.

CL

Dr. Sarat C. Tripathy*

Professor

Centre o f Energy Studies Indian In stitu te o f Technology, Delhi

New Delhi 110 016, India

Dr. G.S.S.S.K. Durga Prasad Lecturer

Department o f E le c tr ic a l Engineering Indian In stitu te o f Technology, Delhi

New Delhi 110 016, India

ACKNOWLEDGEMENTS

I am highly indebted to Prof. S.C. Tripathy and Dr. G. Durgaprasad for their constant guidance and encouragement in pursuing this research work. It has been my privilege to have received their help at various stages of the investigation.

I would like to thank the authorities of the Banaras Hindu Univer

sity, Varanasi, and the Ministry of Education, Government of India, for providing me the QIP scheme facility.

I am thankful to (Late) Prof. A.K. Sinha, Prof. C.S. Indulkar,
*i*

Prof. B.P. 'Singh and Dr. R. Balasubramanian for their valuable discus

sions during the course of this work.

The patience and understanding of my wife, Jyotsna, has been of great source of inspiration and help in the progress of this work.

I wish to thank my colleague Mr. A.K. Wahi and Kail ash for their assistance during the preparation of this thesis.

I would like to thank the Head, Department of Electrical Engineer

ing, Indian Institute of Technology, Delhi, for providing me all faci

lities in the department to carryout my research work.

Finally I would like to thank Mr. P.M. Padmanabhan Nambiar for typing the thesis and Mr. Kapoor for making neat tracings of figures.

DURG SINGH CHAUHAN

ABSTRACT

The th esis presents the resu lts o f research carried, out by the author in the area o f power system sta te estim ation over the period, July 1982 to August 1985, at the Indian In stitu te o f Technology, Delhi, India.

The th esis con sists o f seven chapters o f which the f i r s t one i s in troductory and the l a s t one i s the conclu sion . The review i s described in Chapter I I 'and the re su lts o f th is in v estig a tion are described in Chapters I I I through VI. The b a sic theme o f the inve

stig a tio n i s to develop an algorithm which i s computationally e f f i cie n t and v e r s a tile . S p e c ific a lly , we have u t iliz e d the concept o f the Hessian matrix approach in the framework o f existin g WLS e s t i

mator. A b r i e f d escrip tion o f the content o f chapters i s as follow s **i**

Chapter I s This chapter i s a b r ie f in trodu ctory chapter which attempts to elaborate the importance o f the s ta tic state estim ation in the con trol room operation with s p e c ific features o f ‘the present day estim ation techniques.

Chapter I I s This chapter deals with the d eta iled mathematical model o f estimation algorithms available in the lit e r a t u r e . Standard WLS estim ator, decoupled estimator in polar coordinate and exact second order estimator in cartesian coordinate are f u l l y derived. Bad data suppression estimators i n 1-amped and decoupled form based on non~

quadratic co s t function arc presented.

Chapter I I I : This chapter deals with the Hessian matrix approach incorporated in the w ell known WLS methods in polar as well as cartesian coordinates. Ill-c o n d itio n in g o f information matrix i s ofte n a problem in the weighted le a s t square estimation and i t could be solved with nominal computational e f f o r t using the follow ing methodology.

Using the Hessian matrix approach the inform ation matrix [m] = [m^J [m2] .

where

= the inform ation matrix o f WLS method

[M 1 = the matrix o f second order d eriv a tiv e o f fu nctions.

2

I f we ignore the second order terms, the algorithm becomes u n fit in ill-c o n d itio n e d or bad conditioned case. Proposed Hessian matrix approach does not a f fe c t the spa rsity o f the information matrix used in the WLS method.

Chapter IV s This chepter completely describes the tracking sta te estimation algorithm using the proposed apporach. Bad data suppression algorithm, using the Hessian matrix in decoupled ver

sion i s described. P re filte rin g i s incorporated to f i l t e r out gross measurement errors. A tracking o f states from the nomine!

base case to fin a l operating' con d ition has been done in 40 samples by using a ll the aforesaid techniques. The proposed technique tracks the sta te on the curvature o f the c o st fu nction.

Chapter V s This chapter deals with the analysis o f power system o b se rv a b ility . Making use o f simple measurement graph and i t s spanning tr e e algorithm, c r i t i c a l measurement graph and i t s span

ning tree algorithm, c r i t i c a l measurement algorithm and pseudo measurement placement algorithms are developed. O bservability o f power system i s an e sse n tia l aspect o f processing the measure

ment v e c to r . The proposed algorithm has computational su p eriority over other algorithms available in the lite r a t u r e .

Chapter VI 8 Multiple bad data id e n t ific a t io n using s e n s it iv ity matrix i s a new approach. This chapter describes the use o f new s e n s it iv ity matrix o f the proposed Hessian matrix estimator in finding e f f e c t o f bad data i n the residual vector o f the estimate.

Normalized s e n s itiv ity matrix approach i s exploited to estimate the random noise vector o f suspected bad data, as normalized residual i s b est judged in id e n t ific a t io n o f sin g le bad data.

Bad data e f f e c t spread on normalized residual has lin e a r re la tio n with normalized s e n s itiv ity matrix. I t can give, b est co rre ctio n fa c to r . This theme i s presented in th is chapter with te s t resu lts conducted on a 30-bus w ell-con dition ed power system.

Chapter VII s This chapter i s devoted ex clu siv e ly to comments on achievements o f the in vestiga tion s reported in e a r lie r chapters and future scope o f the research work.

**CONTENTS**

Page

.Abstract i

Chapter I Introduction 1

1.1 General Outline 1

1.2 Features and Fundamentals o f Estimation 2

1.3 S tatic State Estimation 4

1.4 Tracking State Estimation 4

1.5 Power System O bservability 5

1.6 M ultiple Bad Data Id e n tifica tio n 5

1.7 Outline o f the Thesis 5

Chapter I I Review o f S ta tic State Estimation 8

2.1 Introduction 8

2.2 Weighted Least Square Estimator 9

2.3 Problem Formulation 10

2.4 The Weighted Least Square Estimation 13 ALgorithm Development

2.5 Recursive Processing 15

2.5.1 Constant Gain ALgorithm 16

2.6 Decoupled State Estimation 17

2.6.1 Decoupled estim ation with complete 18 data vector

2 .6 .2 Completely decoupled estimator 21 (Fast decoupled estimator)

2.7 Cartesian Coordinate Formulation o f 23 Weighted Least Square (WLS) Method

2.7.1 Fast exact second order sta te {FESOS) 25 estimator

2.8 Bad Data Suppression Estimators 27

2.8.1 The bad data suppression (BDS) algorithm 26

2 .8 .2 Fast decoupled BDS estimator 33

Chapter I I I S ta tic State Estimation Algorithms Using 33 Hessian Matrix .Approach

3.1 Introduction 33

3.2 Development o f ALgorithm 36

3.3 Fast Decoupled State (FDS) Estimator 39

. using Hessian Matrix Approach .

3.4 Fast Decoupled Exact Second Order 40 . Estimator (FDESOS)

3 .4 .1 FDESOS using Hessian Matrix 41

3.5 Convergence C haracteristics 42

3.6 D igital Simulation and Results 43

3.7 Conclusions 64

Chapter IV A Tracking State Estimation in Power Systems 66 '

4.1 Introduction 66

4.2 Formulation o f Problem 67

4.3 Basic WLS Tracking State Estimator using 69 Hessian Matrix

. 4.4 The Fast Decoupled Model using 69

Hessian Matrix

■ 4.5 A P re filte rin g ALgorithm Formulation 71 4 .5 .1 The measurements and the sta te modal 72 4 .5 .2 Loop in cidence matrix formulation 74

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4.6 Fast Decoupled Bad Data Suppression 75 ALgorithm using Hessian Matrix

4.7 Simulation D escription 76

4.8 Numerical Results* 99

4.9 Conclusions 99

Chapter 5T Power System O bservability 100

5*1 Introduction _ 100

5.2 Relationship between the Measurement 102

■ Mode], end -the Topology o f Network

5.2.1 ALgebraical and numerical o b se rv a b ility 104

5 .2 .2 Topological O bservability 105

5.3 The Measurement Graph 106

5.4 The Spanning Tree Formulation ALgorithm 110 5.5 Determination o f C ritica l Measurements 111 5.6 The Pseudo Measurement ELacement ALgorithm 118 5.6.1 ALgebraic concept o f pseudo measurement 120

placement

5.7 D igital Simulation D escription 121

5.8 Conclusions . 132

Chapter VI Multiple Bad Data I d e n tific a tio n

6.1 Introduction 133

5.2 S tatic State Estimation in E le c tr ic 136 Power System - General Formulation and

Notations

6.2.1 S tatic state estimation 136

6 .2 .2 S en sitiv ity analysis 138

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. 6 .2 .4

' 6 .2 .5
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^{2 .6}6 .2 .7
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^{2.8}6.3 6.3.1 6 .3 .2 6.3.3 6.4 6.5 Chapter VII

Appendix 1 .Appendix 2 Appendix 3 .Appendix 4 .Appendix 5 Appendix 6

**3 X**

S e n sitiv ity matrix [---

### J

**3 Z**
3£

S e n sitiv ity matrix [ J 3 Z

Weighted and normalized covariance methods

The r o le o f normalized residuals Detection' theory o f bad data

Some in terestin g properties o f r ^ -te s t and r ^ -te s t

The Estimation Id e n tific a tio n .Approach fo r Id en tify in g M ultiple Bad Data

Mathematical d e scrip tion o f the approach Correction o f the sta te estimates

**Iden tifiabilxty of bad data**

D escription o f Model System ancj Simulation Conclusions

Conclusions

Scope for Future.Research References

Hessian Matrix o f the Functionals Decoupled lies s i an Matrix

Decoupled Hessian Matrix (Rectangular) Proposed BDS Hessian Matrix '

Graph theory

Curriculum Vitae

138

139

140 141 143 144

145 146 150 151 151 163 164 167 168 179:

188 191 19 6 19 8 203 214

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