An Extended Optimal Power Flow Measure for Unsolvable Cases Based on Interior Point Method

8/2/2019 An Extended Optimal Power Flow Measure for Unsolvable Cases Based on Interior Point Method

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An Extended Optimal Power Flow Measure

for Unsolvable Cases Based on Interior

Point Method


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CONTENTS:

1) Introduction.2) Voltage Stability Curve.3) Method of Identifying Weak Lines and Weak Buses.4) Recovering Power flow solvability.5) Extended Optimal Power Flow by Interior Point

Method.6) Different Types of EOPF model.7) Test Results.8) Conclusion.

9) References.


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INTRODUCTION

Voltage stability

Extended Optimal Power Flow(EOPF) model

Assessment of Weak Bus or Line causing Voltage Instability.

Model for Recovering Power Flow Solvability.


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Assessment of VOLTAGEINSTABILITY :

P = Active power at particular bus

V = Voltage at correspondingbus.

Upper half of Curve is stableoperating region

At Nose point Jacobian becomessingular

Voltage Instability occurs when

Reactive power Demand >Reactive

power Generation.


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WEAK BUSES OR CRITICAL BUSES

Buses, which are responsible for voltage instabilitycalled weak buses

Weak buses exhibit following properties

has the lowest reactive power margin

has the greatest reactive power deficiency,

has the highest percentage change in voltage

has the highest voltage collapse point on the V-Q curve,

Weak bus


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DETERMINATION OF WEAK LINES OR BUSES

Method

Solve the power flow equation

If power flow has no solution use optimization methodto solve it

Calculate Extended Line Stability Index (ELSI) for eachline


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CALCULATION OF ELSI.

Figure : Equivalent representation of transmission line

Formula for ELSI


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MODEL FOR RECOVERING POWER FLOWSOLVABILITY

s.t. PGi PDi + Ci - = 0 (2)

QG i + QCri - QDi + CiQDi/PDi -

(3)

eifm - emfi = 0, t = 1,.NT (4)

ei ktem = 0, t = 1,.NT (5)

ktmin kt ktmax t = 1,.NT (6)

PGtmin PGi PGtmax, t = 1,.NG(7)

QGtmin QGiQGtmax t=1,.NG

Qcrimax Qcri Qcrimax, t=1,.Ncr

0 C PDi, t =1,NB

1

min (1)BN

i i

i

w C

Li Ti

Lij Tij

ij S ij S

P P

0,

Li Ti

Lij Tij

ij S ij S

Q Q


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The Extended OPF Model based On Interior pointmethod :

Optimization of OPF

obj.min. f (x)(1)

s.t. h(x) = 0(2)

gl g(x) gu(3)

h(x) -> Nodal power equality constraintsg(x) -> Inequality constraints

With the help of Slack Variables, inequality constraints transformed toequality...

g(x) + u = gu

g(x) l = gl


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Adding Slack Variables to Equality constraints to expand the feasibleregion

h(x) = 0 => h(x) + s = 0 s > 0

Obj min f(x) => obj. min. f(x) + M Where M is a large number calledpenalty

coefficient (4)4

Adding Slack Variables to Inequality constraints to expand the feasible

region,

g(x) + u = gu => g(x) + u = gu + u u>0g(x) - l = gl => g(x) - l = gl - l l> 0

The objective function is :

1

r

jj

s

Adding slack to Equality constraint = Injecting active or Reactivepower

Adding slack to Inequality constraints = loosening the securitycriterions

' '

1 1

min ( ) => min f(x) + M + M (5)r r

j j

j j

f x u l


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Determination Of Key Constraints set :

In the process of optimization, lot of slack variables should be non zero

to restore OPF solvability ,many adjustments should be carried out toobtain an approximate optimal solution.

Extended Optimal Power Flow method is designed to determine thekey

constraint leading to insolvability and hence, the approximate optimalsolution with fewer adjustments to the constraints.

The methods used in EOPF area) Linear Penalty Model.

b) Square Penalty Model.c) Root Penalty Model.


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Comparison of Root Penalty Model and Square Penalty Model

Here Pmax < Pa and Qmax > Qa + QbFollowing two schemes can be used to restore solvability.

Scheme 1 : power injection of node a is & power injection of node b is

+ = Pa + Pb Pmax (6)

Scheme 2: power injection of node b is = Pa + Pb Pmax (7)+ =

ap

bp

bpap

abp

ap bp abp


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Detemining key constraints using Square penalty model:

obj 1 = M ( pa2 + pb

2 ) (8)

obj 2 = M ( pa + pb )2 (9)

obj 1 < obj 2 Excessive adjustment locations

Determining key constraints using Root Penalty Model:

For the Root penalty model in, the value of penalty item in objective in scheme 1 is,

obj 1 = M ( ) and (10)

obj 2 = M = M (11)

obj 1 > obj 2

Optimal solution inclines to Scheme 2, with few adjustment solutions as possible.

a bp p

ab

pa bp p


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Simulation Results :

RTS - 24 system is used to calculate the model with techniques

presented:

RTS 24 system consists of LOAD buses, GEN buses,

Two situation are considered:

A) Infeasibility due to strict nodal voltage amplitude constraints.B) Infeasibility due to constraint of transmitting line power limit in

certain line.

Observations are made on the basis of following:A) Effect of Slack variables into Equality constraints.B) Effect of Slack variables into Inequality constraints.


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Simulation Results forLinear

Penalty Model:

A) Infeasibility due to NodeVoltage constraints

Virtual Reactive Powerinjection

at nodes to optimize the powerflow within constraint limits.

Upper or lower nodal voltageamplitude constraints are

broadened for the model ofintroducing slack variables toinequality constraints.


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B) Infeasibility due totransmission lineconstraint:

Active power injection atseveral nodes can be obtainedfor the model of introducingslack variables to Equality

constraints.

The transmitting limit in thevery line which leads to anunsolvable state can beincreased to obtain a

approximate solution for themodel of inequalityconstraints.

The transmitting limit in line

15 -24 is increased to 1.102


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Results of Square penalty Model v/s Root penalty model

For square penalty model:reactive power injections at about

5nodes are injected to restore the

OPF solvability for square model.

Root penalty model requires far

fewer number of adjustmentlocations. For 0.99-1.01, injection

atone node is needed.


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CONCLUSION

EOPF provides the planning and operation staffs with practicaladjustment scheme.

The Model proposed in this paper can solve the optimizationproblem, requiring fewer adjustment locations, especially forthe root penalty model.

Through the Introduction of slack variables in extended OPFmodel,

the speed of restoring solvability is remarkably improved


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REFERENCES

[1] Lin Liu, Xifan Wang, Xiaoying Ding, Furong Li, Min Fu, An Extended

OptimalPower Flow Measure for Unsolvable Cases Based on Interior PointMethod

IEEE Trans. Power Syst., vol. 16, no. 2, pp. 222228, May 2009.[2] Wei Yan, Juan Yu, David, C.Y., and Kalu Bhattarai, A new optimal reactive

power flow model in rectangular form and its solution by predictor corrector

primal dual interior point method, IEEETransactions on Power Systems,2006,

21, (1), pp. 61-67[3] Wenyuan Li, Juan Yu, Yang Wang, Paul Choudhury, and Jun Sun, Methodand

system for real time identification of voltage stability via identification of

weakest lines and buses contributing to power system collapse. U.S.Patent

7816927, Oct. 2010 (filed Jul. 27, 2007), China Patent Zl200710092710.1,Aug

2009 (filed Sep. 17, 2007).[4] G. L. Torres and V. H. Quintana, An interior-point method for nonlinear

optimal power flow using voltage rectangular coordinates [J], IEEET

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An Extended Optimal Power Flow Measure for Unsolvable Cases Based on Interior Point Method

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