State Estimation in Power Distribution Network Operation Ravindra - - PowerPoint PPT Presentation

State Estimation in Power Distribution Network Operation

Ravindra Singh

• Dept. of Electrical & Electronic Engg.

Imperial College London

1

Outline:

• Introduction
• Research Objectives

– Identification of DSSE solver – Modelling of pseudo measurements – Measurement placement

• Conclusion and future Work

2

Introduction

Distribution networks of today:

– Old design and ageing equipments – Passive operation – Based on “Fit and Forget” concept

This concept can not work considering

– The deregulation of power system – Emphasis towards low carbon technology – Introduction of DGs at distribution level

3

Distribution Management System (DMS)

4

• Introduction of DGs at distribution level requires monitoring and

control of distribution networks through Distribution Management System (DMS)

• DMS has two functional block
• State Estimation
• Control Scheduling
• SE will be the core function of the DMS which will provide the input

to control scheduling block for various control functionality

Challenges in DSSE

• State estimation is routine task in

transmission systems

• Several SE techniques exist
• But these techniques can not be

duplicated in distribution networks mainly due to

– lack of available measurements – lack of methodologies and tools that can be applied to restricted measurements

6

Objectives

1. Development of DSSE Algorithm

• Choice of DSSE Solver

2. Modelling of Pseudo Measurements

• To ensure network observability
• To handle non Gaussian load distribution

3. Measurement placement

• Location
• Type
• Number

4. SE with network topology changes

7

Objective 1: Choice of DSSE Estimator

• State estimation is a data fitting

problem:

– State Vector: – Measurements: – Model:

• The solution of the state

estimation problem requires finding that minimises a (weighted) norm of the residuals:

฀ ฀ zi ∈ ℜ, i =1,K ,m

n

x ℜ ∈

฀ ฀ zi = hi(x) + ei, i =1,K ,m

m i x h z x r

i i i

, , 1 ), ( ) (  = − =

x

m > n, ei = measurement error (Gaussian)

8

Review of state estimation in transmission system

L2 and L1 norms

• L2 norm:

(Weighted) Least Squares LS approach.

– Performs well in the presence

• f noise

– Fails to reject bad data

• L1 norm:

(Weighted) Least Absolute Value LAV approach

– More robust than LS in the presence of bad data – Ineffective for noise filtering min

x

1 2 wiri(x)2

i=1 m

∑

where wi = 1 σ i

2

min

x

wi

12 ri(x) i=1 m

∑

9

Compromise: Huber M-estimator

• Huber function
• State estimation

problem ρ(t) =

1 2 t 2, if t ≤ γ

γ t − 1

2 γ 2, if t > γ

  

∑

= m i i x

x r

1

)) ( ( min ρ

estimator LS ⇒ ∞ → γ estimator LAV 0 ⇒ → γ

10

Estimators Summary

ϒ= 1.5

11

H = ∂h(x) ∂x and Rz is measurement error covarince matrix

• Various estimators like WLS,

WLAV, and SHGM were tested on the 95 bus UKGDS model (Fig) with

• errors in real measurement

1%-3%

• errors in pseudo measurements

20%-50%

• DSSE relies on a large number of

pseudo measurements which are statistical in nature

• Thus the statistical criterion were

adopted to identify the suitable solver for the DSSE problem

12

Simulation study

Statistical Criteria

• Bias: For the estimator to be unbiased, the mean error in

the estimates should be zero

• Consistency : For consistency the normalized mean

square estimation error should be within confidence bound determined by chi-square test (under Gaussian assumption

• f error)
• Quality : Expressed as the inverse of the trace of the error

covariance matrix

13

E( ˆ x − xt) = 0

ε = ( ˆ x − xt)T P −1( ˆ x − xt)

Q = 1 log(trace(P))

Performance Summary

14 8,86

• 55,65

6,7

• 60
• 50
• 40
• 30
• 20
• 10

10 20 WLS WLAV SHGM

Bias Consistency Quality WLS is the suitable solver for the DSSE problem

Objective 2: Modelling of Pseudo Measurements

5000 10000 15000 0.5 1 1.5 time Load,pu 5000 10000 15000 0.5 1 1.5 time Load,pu 5000 10000 15000 0.5 1 1.5 time Load,pu 5000 10000 15000 0.5 1 1.5 time Load,pu Commercial Domestic Economy Domestic Unrestricted Industrial

Loads of various customer class Load profile computations

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WLS algorithm assumes that the measurements are normally distributed, however:

• Load pdfs do not follow normal distribution
• One way could be to model them as lognormal or beta distribution
• However, this is not true (pdf shown in Fig.)
• This shows that the load pdfs do not follow any particular known

distribution

Issues with the WLS

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How to Model the Variability in Load pdfs?

Concept of Gaussian Mixture Model (GMM)

• Load pdfs are represented

through a mixture of Gaussian components

• Resulting pdf is a weighted

combination of individual Gaussian components.

• Now WLS can be applied
• Use of Expectation

Maximization (EM) algorithm to obtain the parameters of the mixture components

17

Density Estimation Through EM Algorithm

{ }

c o m p o n e n t s m i x t u r e

• f

N

• .

) d e t ( ) 2 ( 1 ) , | ( } , , { : s e t t h e f r o m c h o s e n i s w h e r e , 1 a n d ) , | ( ) | (

) ( ) ( 2 1 2 / 1 2 / 1 1 1

1

= Σ = Σ Σ = = Γ = Σ =

− Σ − − = = =

−

∑ ∑

M c e z f w w z f w z f

i i i

z z i d i i M c i i i i M c i i M c i i i i µ µ

π µ µ γ γ γ µ γ

• Mixture Model

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The Algorithm

The EM Algorithm works recursively to obtain the parameters of the mixture

• components. One step of recursion, yielding formulae for :

i g i v

1 s s

γ γ

+

Result at Bus #82

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Comparison with other distributions

• Comparison is based on both graphical

and the numerical measure

• For numerical measure Chi-Square

goodness of fit value is used

• A smaller value of goodness fit

indicates a better fit

• GMM gives better fit according to

both measures

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SE Results:

• SE was performed with 1 voltage and 2

flow measurements at main substation

• Loads were modelled as GMM using

load profiles obtained by mapping the behaviour of various customer class at each bus

• SE was run at different time steps by

sampling the load profiles

• Estimation errors are small close to

the main substation buses (Bus #2)

• Errors increase at the buses away

from main substation (Bus #51)

• This is because the buses away from

main substation are predominantly influenced by pseudo measurements

• Error in angle is more as compared to

voltage

Bus #2 Bus #51

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Objective 3: Measurement Placement

SE Example:

• 3% error in real, 50% error in pseudo
• UKGDS Network
• 100 Simulations
• 1% voltage threshold 5% angle

threshold

• In this example a significant number of cases are above their respective thresholds
• The reason is that the SE utilizes a large number of pseudo measurements
• This problem can be eliminated by placing more number of real measurements

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Objective

To bring down the relative errors in Voltage and angle below threshold in more than 95% simulation cases.

Benefit

Reduction in error in estimates.

Issues: To identify

• Measurement location
• Measurement type
• Number of measurements

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Problem Formulation

Improve: In order to improve above index reduce: Where, (1) (2)

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Geometric Interpretation of Error Covariance Matrix

n-dimensional error ellipse e = error vector (n×1) P= error covariance matrix (n×n) c=constant Using transformation , can be diagonalized and the is given by:

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• Index in (1) can be improved by reducing

the right hand side of (2)

• This can be achieved by reducing the

determinant of estimation error covariance matrix.

• Since determinant represents the area of

the error ellipse, the said probability index can be improved through reduction in area of the V-δ error ellipse.

• Placement of voltage meter reduces the

errors in both voltage and angle if

• rientation of ellipse is according to Fig.

(a) (Correlated)

• However if the errors are aligned with the

coordinate axis (Fig. (b)). The placement

• f voltage measurement does not

improve errors in angle (Uncorrelated)

• In this situation we used flow

measurements and based on the criterion

• f minimization of the area of P-Q_flow

error ellipse

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The Algorithm Flow Chart

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Results

Measurements Placed: V19,V20, V21 P15-17,Q15-17 P34-35,Q34-35

Quality Error ellipses

Red ellipse: Before No Placement Blue ellipse: After All Meters Placed

• 2

2 x 10

• 3
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 Error in Angle Error in Voltage

• 2

2 x 10

• 3
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015

• 2

2 x 10

• 3
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 Bus #19 Bus #20 Bus #21

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Chebyshev Bound

• Minimization of probability in (2)

makes sense if Chebyshev bound is less than unity

• This depends on the choice of

the threshold and increases with reduction in threshold

• A very tight threshold may not

be useful for practical purpose hence algorithm can be started with appropriate choice of thresholds

• In measurement placement

algorithm bound reduces significantly with placement of meters

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Conclusion & Future Work

• Problems associated with DSSE have

been highlighted and addressed in probabilistic and statistical framework

• Future work will focus on

– Three phase SE – SE with correlations in measurements – Decentralized estimation – Stochastic optimization based measurement placement technique

» T

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Publications

• Journal

[1] R. Singh, B. C. Pal, and R. B. Vinter, “Measurement Placement in Distribution System State Estimation,” IEEE Transactions on Power Systems, vol. 24, no. 2, pp. 668-675, May 2009 [2] R. Singh, B. C. Pal, and R. A. Jabr, “Choice of Estimator for Distribution System State Estimation,” IET Generation Transmission and Distribution, vol. 3, no. 7, pp. 666-678, July 2009 [3] R. Singh, B. C. Pal, and R. A. Jabr, “Statistical Representation of Distribution System Loads Using Gaussian Mixture Model,” IEEE Transactions on Power Systems, Accepted for publication. [4] R. Singh, B. C. Pal, and R. A. Jabr, “Distribution System State Estimation Through Gaussian Mixture Model of the Load as Pseudo Measurement,” IET Generation Transmission and Distribution, Revision under review. [5] R. Singh, E. Manitsas, B. C. Pal, and G. Strbac, “A Recursive Bayesian Approach for Identification of Network Configuration Changes in Distribution System State Estimation,” IEEE Transactions on Power Systems, Revision under review.

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• Conference

[6] E. Manitsas, R. Singh, B. C. Pal, and G. Strbac, “Modelling of Pseudo Measurements for Distribution System State Estimation,” CIRED, 2008, Frankfurt. [7] R. Singh, B. C. Pal, R. A. Jabr, and P. D. Lang, “Distribution System Load Flow Using Primal Dual Interior Point Method,” IEEE Powercon and Power India Conference, New Delhi, 2008. [8] R. Singh, B. C. Pal, and R. B. Vinter, “Measurement Placement in Distribution System State Estimation,” IEEE PES General Meeting, Calgary, July, 2009.

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State Estimation in Power Distribution Network Operation

Ravindra Singh

• Dept. of Electrical & Electronic Engg.

Imperial College London

40