An Extended Optimal Power Flow Measure

8/2/2019 An Extended Optimal Power Flow Measure

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Chapter 1

Voltage Stability

1.1 Introduction

Voltage stability is the ability of a power system to maintain steady acceptable voltages at all

buses in the system under all operating conditions. Voltage control and stability problems are

now receiving special attention , as under heavy loaded conditions there state may be insufficient

reactive power causing the voltages to drop. This drop may lead to drops in voltage at buses.

This sort of abnormal voltage drop is referred as Voltage Instability. In some classical studies,

voltage magnitude related problems were viewed as local phenomena. Voltage stability though is

essentially a local phenomenon, but its consequences may have widespread impact. In recent

years, with lines being overloaded all other reasons tending to voltage instability, analysis of the

system for voltage stability may be done to understand about the reliability of system and

magnitude at a bus. Maximum loadability limits of a load bus are also determined in terms of

voltage stability.

1.2. Analysis of Voltage Stability (PV curve)

The PV curve is a power voltage relationship at receiving bus. Figure 1.1 is an illustration

of a typical PV diagram. V in the vertical axis represents the voltage at a particular bus while P

in the horizontal axis denotes the real power at the corresponding bus or an area of our interest.

The solid horizontal nose-shaped curve is the network PV curve while the dotted parabolic curve

is the load PV curve. The operating point is the intersection between the load and the network

curves . Load PV curve shows the variation of power consumed by a load at a bus with respect tovoltage applied to the load which depends upon the load characteristics. The commonly referred

PV curve is the network PV curve. It is the network voltage response at a particular bus due to

load increase in a certain area or bus of a power system, as the system moves from one operating

point to another, constant power characteristics and power factor of the load is assumed. The top

half of the curve is the stable solution while the bottom half is unstable (determined by load


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characteristics but deemed unfeasible for power system operation due to high current and low

voltage). The two solutions coalesce at a point called the critical point (also referred as, the nose

point or the point of maximum power transfer). Beyond this point, the power flow does not

converge. There are number of factors such as the generator reactive power limit, contingences,

load dynamics, stress direction, etc that affect the distance of the nose point from the point of

operation. By understanding these factors the system can be steered away from the nose point

and make the system

Figure: 1.1 Load and PV curves


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1.2.1. PV Curve Tracing:

PV curve tracing is computationally intensive and requires proper techniques to avoid

numerical instability. For a simple two bus system, a closed form expression can be developed.

A series of network PV curves (for varying power factor) has been drawn using this expression

in Figure 1.2. Although the curves are for a two bus system, the shapes are quite general.

A closed form expression for voltage and power in large systems (systems with more than two

buses) is not possible. In such a case, the technique is to solve the power flow equations

numerically for each operating point. This makes the tracing highly computational. As the

system gets closer to the nose point, getting convergence is difficult. This is because, the power

flow Jacobian approaches singularity towards the nose point and becomes singular when it is at

the nose point. The singularity causes the power flow solution to diverge. Continuation power

flow (CPF) method is commonly used to solve the divergence problem.

Figure 1.2 PV curve for different power factors


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Chapter 2

Method of identifying Weak lines and Buses causing Static Voltage

Instability:

2.1 Line Based Voltage Stability Index

It can be seen from the definitions of the risk indices that the key step towards calculating

the risk indices is identification of weak lines and buses causing static voltage instability, i.e.,

determination of the three numbers mWBi, mWLij and m in Monte Carlo simulation. Reference [3]

presented a line-based voltage stability index for real time application, which is called the

Extended Line Stability Index (ELSI). The ELSI is calculated

2 * 2 2 2 * 2/ 2[ ( )( ( ) ) (1)k kj ij kj ij ij ij i j ijELSI E R P X Q R X P Q

where Rkj+jXkj is the equivalent line impedance of a line in which the effect of equivalent

voltage source outside the two buses of the line has been incorporated; P+jQ*ij is line complex

power flow with the charging reactive power excluded at the receiving bus j; and Ek is the

voltage of the equivalent voltage source outside the two buses of the line and the formula of

calculating Ek using bus voltages was derived in Reference [3].

The ELSI of all lines must be larger than 1.0 in order to keep system voltage stability. When the

ELSI of a line is approaching to 1.0, the line and its receiving bus become weak. Once the ELSI

of at least one line is sufficiently close to 1.0, the system reaches the critical voltage instability

point. The details of derivation, proof and tests can be found in Reference [3].

The only localized information (line impedance, power flows and voltages at two ends of a line)

is needed to calculate the ELSI. In real time application, the localized information is directly

obtained from Phasor Measurement Units (PMU). In the off-line application of calculating the


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proposed risk indices, a considerable number of power flows which are sampled in Monte Carlo

simulation, are solved in order to provide the information required for calculating the

ELSI of each potential weak line in various system states. It should be pointed out that not only

the ELSI but any other localized line-based or bus-based voltage stability index can be also used

in the presented method as long as it can identify system voltage stability and weak lines and/or

buses.

2.2 Model for Recovering Critical Power Flow Solvability

As mentioned above, in real time application of the ELSI, no power flow is needed since the

localized information can be directly obtained from PMUs. However, power flows of various

system states which are randomly selected in Monte Carlo simulation need to be solved in order

to calculate the ELSI. If a power flow is unsolvable, the ELSI or any other line-based or bus-

based voltage stability index cannot be calculated to identify system voltage instability and weak

lines/buses. There are two possibilities for a case of no power flow solution. One possibility is

that the system is not voltage stable. The other one is that the system is voltage stable but a

numerical instability problem occurs in power flow calculations. Therefore the system voltage

stability cannot be accurately identified only from solvability of power flow. However, once the

system is recovered to a critical solvable state, the line-based index ELSI can be used to make adifferentiation between the two possibilities. If the ELSI is very near 1.0 in the recovered critical

state, the prior system state before recovery is a voltage instable state; otherwise, if the ELSI is

quite larger than 1.0, no power flow solution for the prior system state is caused due to a

numerical stability problem.

The optimal model, as expressed in Equation (2)-(11), is presented to recover the critical

solvability of an unsolvable power flow through control variable optimization and minimum load

shedding. The main advantage of this model is its quadratic form. This leads to a constant

Hessian matrix which needs to be calculated only once in the entire optimization process. This

feature significantly decreases the computation time and complexity in resolution. The predictor

corrector primal dual interior point method (PCPDIPM) and the sparse technique [2] are used to

solve the model.


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In the model, the real and imaginary parts ei and fi of bus voltage at Bus i, turn ratio kt of the tth

on-load-tap-changer (OLTC) transformer branch, active and reactive power injections PGi and

QGi of generators at Bus i, reactive power injections Qcri of shunt reactive compensator and

active load curtailment Ci at bus i are unknown controllable variables to be optimized, whereas

active and reactive loads PDi and QDi at Bus i are known quantities.

1

min (1)BN

i i

i

w C

s.t. PGiPDi + CiLi Ti

Lij Tij

ij S ij S

P P

= 0 , i = 1,2..NB (3)

QG i + QCri - QDi + CiQDi/PDi - 0,Li Ti

Lij Tij

ij S ij S

Q Q

(4)

eifm - emfi = 0, t = 1,.NT (5)

ei ktem = 0, t = 1,.NT (6)

ktmin kt ktmax t = 1,.NT (7)

PGtmin PGi PGtmax, t = 1,.NG (8)

QGtmin QGiQGtmax t = 1,.NG (9)

Qcri min Qcri Qcri max, t = 1,.Ncr (10)

0 C PDi, t = 1,NB (11)

where wi is the weighting factor reflecting load importance; Equations (3) and (4) are the active

and reactive power balance constraints at buses; the reactive load at Bus i may be shed with

active load curtailments in terms of constant power factor, which is expressed in Equation (4);NB, NG, Ncr and NT denote the numbers of system buses, generator buses, reactive compensation

buses and OLTC branches respectively; Gij+jBij is the ith row and jth column element of the bus

admittance matrix in which the contributions of all OLTC branches have been excluded; the

quantities with subscript max or min represent the maximum and minimum limits of

corresponding variables. The STi is the set of all the OLTC branches connected to Bus i. The SLi


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is the set of all the lines and non OLTC transformer branches connected to Bus i. The quadratic

power expressions of OLTC branch ij are given by Equations (12)-(15) and illustrated in Figure

2.1, where m and gt+jbt represent a dummy node and the branch admittance respectively. The P Tij

in Equation (3) and QTij in Equation (8) are given by Equations (12) and (13) respectively when

bus i is located at the high voltage side of OLTC; otherwise. Equations (5) and (6) denote the

voltage conversion relation of OLTC branches.

PTij = (em2+ fm

2- emej - fmfj) gt + (emfjejfm)bt (12)

QTij = - (em2+ fm

2- emej - fmfj ) bt + (emfjejfm)gt (13)

2.3 Identifying Weak Lines and Buses Causing Voltage Instability

Although the proposed optimal model can recover the critical solvability of power flow

through control variable optimization and minimization of load shedding when the power flow is

unsolvable, it does not have an ability of identifying weak lines and buses causing voltage

instability. Hence, the optimization model is combined with the ELSI to recognize system

voltage instability as well as weak lines and buses. The procedure of the proposed method is as

follows:

Step 1: Calculate the power flow of a sampled system state. If the power flow is solvable, the

system state is stable, which makes no contribution to the risk indices. Otherwise, go to Step 2.

Step 2: Use the PCPDIPM to solve the optimization model of the sampled system state as shown

in Equations (2)-(11). If the resulting load curtailment is 0, the corresponding power flow

Fig 2.1 The OLTC transformer model with a dummy node


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embedded in the model has a solution and the sampled system state is stable, which makes no

contribution to the risk indices. Otherwise, if there is a non-zero load curtailment, it

indicates that the corresponding power flow has no solution but a critical solution is recovered

through control variables optimization and minimization of load curtailment; go to Step 3.

Step 3: Calculate the ELSI indices of all lines in the critical system state in which the power flow

solvability is just recovered. If the ELSI indices of all the lines are larger than the threshold

which should be a value larger than but very close to 1.0 (1.01 is used in the

simulations of this paper), the original power flow unsolvability does not result from static

voltage instability but is due to a numerical instability problem. This system state also makes no

contribution to the risk indices. Otherwise, if the ELSI of at least one line falls below the

threshold, the power flow unsolvability is caused by static voltage instability. At the same time,

the lines whose ELSI indices are smaller than the threshold are identified as weak lines, and their

receiving buses are identified as weak buses.

The proposed method for static voltage stability identification has the following

advantages:

It can provide the amount and location of minimum load curtailments that are required to

restore the static voltage stability in a system state when the system voltage stability

limit is exceeded.

It can not only identify system static voltage instability but also locate weak lines and

weak buses causing voltage instability accurately.

The constraints of control variables associated with operational conditions of voltage

stability can be considered, such as the bound limits of active and reactive generation power

injections, turn ratios of OLTC transformers and reactive power injections of reactive

compensation devices. It is also flexible to consider only parts of the constraints if necessary.


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Chapter 3

A New Optimal Reactive Power Flow Model in Rectangular Form

3.1 Introduction

A new optimal reactive power flow (ORPF) model in rectangular form is proposed in this

paper. In this model, the load tap changing (LTC) transformer branch is represented by an ideal

transformer and its series impedance with a dummy node located between them. The voltages of

the two sides of the ideal transformer are then used to replace the turn ratio of the LTC so that

the ORPF model becomes quadratic. The Hessian matrices in this model are constants and need

to be calculated only once in the entire optimal process, which speed up the calculation greatly.

The solution of the ORPF problem by the predictor corrector primal dual interior point method is

described in this paper. Two separate prototypes for the new and the conventional methods are

developed in MATLAB in order to compare the performances. The results obtained from the

implemented seven test systems ranging from 14 to 1338 buses indicate that the proposed

method achieves a superior performance than the conventional rectangular coordinate-based

ORPF.

Optimal Reactive Power Flow(ORPF) : In an Interconnected system, Real and Reactive power of

each plant is scheduled to vary within certain limits in such a way to minimize the operating cost,

meeting a particular load demand.

PCPDIPM :

In Recent years, the predictor corrector primal dual interior point method (PCPDIPM) has been

extensively applied to solve large-sized optimal reactive power flow (ORPF) problems due to its

faster calculation speed and robustness, etc.

The conventional ORPF model in polar coordinates is a higher order problem. Its Hessian

matrices are not constants. So the performance of PCPDIPM for solving the ORPF

problem will be affected. The alternative approach is a rectangular coordinate-based ORPF

model, which represents the ORPF problem in the quadratic functions. The properties of this

approach are described in [3] as:

1) its Hessian is a constant;

2) its Taylor expansion terminates at the second-order term


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without truncation error; and

3) the higher order terms are easily evaluated.

Such quadratic features allow for efficient matrix setup and inexpensive incorporation of higher

order information in a predictor corrector procedure that reduces the number of IPM iterations

for the convergence. Although the voltages in rectangular coordinates are used in [4], the optimal

power flow (OPF) formulation is not completely quadratic because of the presence of tap ratio

variables in the load tap changing (LTC) branch power equations. A fully quadratic formulation

of OPF is proposed in [5]. In that paper, the authors used the current and voltage equations to

establish the OPF model in a rectangular form. However, the number of constraints and variables

increased so significantly that the advantages of the quadratic model were overwhelmed by the

longer time needed for the solution of the higher dimensional system of equations.

In this paper, the LTC branch is represented by an ideal transformer and its series impedance

with a dummy node located between them. The voltages of the two sides of the ideal transformer

can then replace the tap ratio of LTC to express the branch power. Thus, a new quadratic model

for the ORPF problem in a rectangular coordinate is developed. Although the introduction of the

dummy nodes will still result in an increase in the number of constraints and variables of the

ORPF, this increase is much less in comparison to that in [5]. The test results demonstrate that

the emergence of a constant Hessian in the proposed ORPF model greatly reduces the total

execution time of the PCPDIPM solution.


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3.2. MATHEMATICAL MODEL OF ORPF IN THE RECTANGULAR

COORDINATES

The transformer branch with LTC can be modeled as an ideal transformer in series with

impedance, as shown in Fig. 1, where kt is the transformer turns ratio, and yt the branchadmittance.

We can obtain the equivalent circuit for this LTC branch as shown in Fig. 2.

Then, the branch powers and the power losses can be written as :

Yt = gt + jbt ; STij = PTij + jQTij ; STij = PTij + jQTij ; Vi = ei + jfi

Vj = ej+jfj (1)

Fig 3.3 Ideal transformer circuit with dummy node

Fig. 3.2 equivalent circuit of transformer branch


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2 2

2

2 2

2

2 2

( )( )( ) (2)

( ) ( ) ( ) (3)

( ) (

t i j i jt i i t Tij i j j i

t t t

t t tTij i j i j i i i j j i

t t t

tTji t j j i

t

g e e f f g e f bP e f e f

k k k

b b gQ e e f f e f e f e f

k k k

gP g e f ek

2 2

) ( ) (4)

( ) ( ) ( ) (5)

tj i j i j j i

t

t t

Tji t j j i j i j i j j i

t t

be f f e f e f k

b gQ b e f e e f f e f e f

k k

From (2)(5), it can be seen that the branch power equations of the LTC branch in rectangular

coordinate system are no longer quadratic because of the tap ratio variable . Even though the rest

of the branch power equations are still quadratic, the higher order terms above will diminish the

advantages of the\rectangular-based ORPF.

In the proposed formulation, a dummy node is added between the ideal transformer and the

series impedance, as shown in Fig. 3.3. The voltage of the dummy node and the branch power

from the dummy node to the impedance are introduced to describe the relationships between the

voltages and the branch powers associated with the LTC branch. For the ideal transformer,

there are no power losses in between its two terminal nodes and ; the ratio of the nodal voltagemagnitudes is equal to the transformer turns ratio; and the nodal voltage angles of

both the nodes are equal. These relationships are described in the following:

,

2 2

2 2

(6)

(7)

( )

Tij Tmj Tij Tmj

i i

t

m m

i

i

P P Q Q

e fk

e f

f

arctg arce

( ) (8)

m

m

f

tg e

Now, the branch flow between and can be modeled as a regular line flow. As (6) represents a

lossless ideal transformer between and , the branch power equations of the LTC transformer

can then be modified as follows:


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2 2

2 2

( ) (9)

+(e f - e f )b

( ) (10)

Tij Tmj

m m m j m j t

m j j m t

Tij Tmj

m m m j m j t

P P

e f e e f f g

Q Q

e f e e f f b

2 2

2 2

+(e f - e f )g

( ) (11)

+(e f - e f )b

( )

m j j m t

Tji Tjm

j j m j m j t

j m m j t

Tji Tjm

j j m j m j t

P P

e f e e f f g

Q Q

e f e e f f b

2 2 2 2

(12)

+(e f - e f )g

= ( 2 2 ) (13)

j m m j t

Tij Tmj

m m j j m j m j t

P P

e f e f e e f f g

From above equations, it can be seen that the branch power flow equations of the LTCtransformer become quadratic similar to the general impedance branches. First two Equations

show the branch power flow from the high voltage side of the transformer to the low voltage

side, whereas next two equations show the branch power flow from the low voltage side of the

transformer to the high voltage side. There are no losses in between the high voltage node and

the dummy node.

The nodal power equations can be written as in (2) and (3). SLi is the set of all general branches

connected to node , STi is the set of all the LTC branches connected to node , NB is the number

of original system nodes, Pi & Qi and are the bus active and reactive power injections.

(14)

(15)

Li Ti

Li Ti

i Lij Tij

j S j S

i Lij Tij

j S j S

P P P

Q Q Q


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The new Nodal Power equations can be written as:

1

1

[ ( ) ( )] (16)

[ ( ) ( )] (17)

B

Ti

B

Ti

N

i ij i j i j ij i j i j Tij

j j S

N

i ij i j i j ij i j i j Tij

j j S

P G e e f f B f e e f P

Q G f e e f B e e f f Q

Where Gij is the ith row and jth column element of the bus conductance matrix , B ij is the ith row

and jth column element of the bus susceptance matrix B, excluding the LTC branches. P Tij and

QTij are the LTC branch powers connected to node . When node is the high voltage side of LTC,

PTij and QTij described by using . When node is the low voltage side of LTC, P Tij and QTij are

described, where and in these equations are switched. Based on above equations ,a new quadratic

model of ORPF is proposed as shown in (3)(12). NG,Ncr , and NT denote the number of the

generator nodes, the reactive compensation nodes, and the LTC branches, respectively;Ns is the

swing bus;


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min ( ) (18)

. .

( )

i=1 ....

Ns

Pi Gi Di i

B

f x P

s t

g x P P P

N

(19)

( )

i=1 .... (20)

g ( ) 0

1.....

Qi Gi cri Di i

B

j i m m i

T

g x Q Q Q Q

N

x e f e f

j N

2 2 2 2 2

max

(21)

0 ( ) ( ) ( )

1......

kj t m m i i

T

h x k e f e f

j N

2 2 2 2

2 2 2 2min ( ) max

(22)

0 ( ) ( ) ( )

1...... (23)

V

1.....

kj i i m m

T

i Vi x i i i

B

h x e f e f

j N

h e f V

i N

min max

(24)

Q ( )

1.....

Gi gi Gi Gi

G

h x Q Q

i N

min max

(25)

Q ( )

1..... (26)

cri ci cri crih x Q Q

i Ncr

The objective function in (18), PNs is the active power injection at the swing bus. The bus active

and reactive power balance constraints are described in (19) and (20). Equations (21)(23) deal

with the LTC parameters. They replace the transformer turns ratio, kt in the conventional ORPF

in rectangular coordinates. This is the key feature of the proposed model. Equation (21)

illustrates that the voltage angles are identical between the high voltage node and the dummy

node. Equations (22) and (23) are the bound constraints of the transformer turns ratio. .Equations

(24)(26) represent the bus voltage, the generator, and reactive compensator bound constraints.

Due to the introduction of the dummy nodes into the system, all the equations in the proposed

ORPF model turn into quadratic. This change will result in constant Hessian matrices in the

model and simplify the computation of the Jacobian matrices. However, there are also some

drawbacks in the model. By replacing the turn ratio with the voltages of the dummy nodes, the

number of constraints and the variables will increase by in the proposed model. The tap ratio

limit becomes a quadratic constraint, in contrast to the simple bound limit in the conventional


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model. Consequently, there is an increase in the number of Lagrange multipliers in the

PCPDIPM algorithm, which, in turn, needs more time for solving the larger Newton systems in

every iteration. Nevertheless, the results shown in the later sections demonstrate that the time

saved in dealing with the constant Hessian matrices is more than the time increased in solving

the Newton system.


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Chapter 4

Extended Optimal Power Flow Measure Based in Interior Point

Method

4.1 INTRODUCTION

For large-scale power systems with heavy load, OPF may not have solutions due to several

reasons, such as strict restrictions of bus voltages, power limits of transmission lines and so on.

Then, system planning and dispatching staffs can merely check the calculation data relying on

experiences or adjust operation mode and plan repeatedly. However, this conventional adjusting

approach is low in efficiency and great in work load. Consequently, it is urgent to find methods

to detect the key constraints that lead to insolvability and adjust the original problem to get an

approximate solution quickly and efficiently.

In this chapter, an extended OPF model based on optimization is discussed. This model is

realized through adding slacking variables to equality and inequality constraints, introducing

penalty items to objective function. Recently, interior point method [7] has been widely used to

solve OPF as it has fast convergence characteristic and can deal with inequalities conveniently.

Improved primal-dual interior method is used to solve the model.

The main features of proposed method are as follows:

1) If the original problem is solvable, optimal solution of original problem can be obtained; if the

original problem is unsolvable due to the violations of constraints or control variables, the

unsolvable OPF can search for optimum in an expanded region automatically and get an

approximate solution quickly.

2) The approximate solution can reflect the key constraints that lead to the insolvability of

original problem and provide adjustive measures clearly.3) The impact of inequality constraints on solvability of OPF is considered. If the solvability can

not be restored under current constraints, the security constraint indices can be loosened if

necessary to get an approximate solution.

4) The improved algorithm has a good convergent property under various conditions. The

proposed method has an advantage over other methods and can be applied to several aspects.


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4.2. An Extended Optimal Power Flow based on Interior Point Method.

In this paper, constraints are slacked through adding slack variables; the extended model

of OPF is obtained through introducing penalty items to objective function. The addition of slack

variables and penalty items could ensure that: if the original problem is solvable, the slack

variables are zero, thus it can converge to original solution; if it is unsolvable, slack variables are

not all zero, thus solutions can be searched in an expanded region, meantime explicit adjustments

can be obtained through slack variables. Specifically, there are two methods:

(1) Slack the inequality constraints by adding slack variables. In addition, penalty items are

introduced in objective function.

(2) Slacking the equality constraints by adding slack variables, and penalty items are introduced

in objective function.

4.2.1 Solving OPF by Primal-Dual Interior Point Method:

OPF can be simplified to the nonlinear optimization model below:

obj.min. f (x) (1)

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

g g(x) g (3)

Where: (1) is the objective function, which represents the fuel cost of generation;

h(x)=[ h1(x) hm(x) ]T L in (2) denotes the nodal power equality constraints;

g (x) = [g1( x).. gr( x)] L in (3) denotes the inequality constraints, including generator output

limit, bus voltage limit and transmission line power limit;

1

1

[ ,...... ] the upper limit and

[ ,...... ] the lower limit.

T

r

T

r

g g g is

g g g is

.

The basic procedure of solving OPF problem with interior point method is described below [16]:

Firstly the inequality constraints are transformed into equality one


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( ) (4)

( ) (5)

g x u g

g x l g

where the slack variables l = [l1,lr]T u = [u1..ur ]T u>0, l>0

Then the Lagrange function is obtained :

L = f(x) - y h(x) - zT [ g(x) lg ]wT [ g(x) + ug ] , where y>0, z>0, w>0 are all lagrange

multipliers.

4.2.2 Adding slack variables to upper and lower limits of inequality

constraints to expand the feasible region

This method is realized as follows :

( )

( )

g x u g

g x l g

=>

( ) ' u' > 0 (6)

( ) ' l' > 0 (7)

g x u g u

g x l g l

The objective function is

Min f(x ) => min f(x) +1 1

' 'r r

j j

j j

M u M l

(8)

Where M is a large positive number. It should be noted that M can also be a vector. If M

is a constant, all of the slack variables adopt the same penalty coefficient; if M is a vector, slack

variables can adopt different penalty coefficients respectively.

Thus it can be seen that the introduction of slack variables and penalty items does not

increase the dimension of correcting equations, therefore it does not increase the computation in

each iteration.


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4.2.3 Adding slack variables to equality constraints to expand the feasible

region

It is realized as follows:

h(x) = 0h(x)+ s = 0 s 0 (9)

obj min f(x) = > obj min f(x) + M1

r

j

j

s

(10)

where M can be a large positive number or vector as in section(previous)

introduction of penalty items does not increase the dimension of correction equations,

therefore it does not increase the computation in each iteration.

4.2.4 Determination of key constraints set

The locations of slack variables and formations of penalty items in the extended OPF

model decide the adjustments to restore the OPF solvability. Adding slack variables in equality

constraints is equivalent to adding virtual active or reactive power in corresponding nodes, while

adding slack variables in inequality constraints is equivalent to loosening the security

criterions.

Under some circumstances, if lots of slack variables should be nonzero to restore OPF

solvability, many adjustments should be carried out to obtain an approximate optimal

solution through slacking many constraints. Then, it can not make clear the key constraints that

lead to insolvability; and it is a disadvantage to practical operation. Consequently, the extendedmodel of OPF should be designed to gain the approximate optimal solution with fewer

adjustments to the constraints, with the purpose of making clear the critical constraints, and

providing the plan and operation staffs with practical adjustment scheme.


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Based on the above concept, the method below is used:

The formation of slacking the equality constraints:

1

min f(x) => obj min f(x) + Mr

j

j

obj s

(11)

The formation of slacking the inequality constraints

1 1

min f(x) => obj min f(x) + M + M (12)r r

j j

j j

obj u l

The models in (11),(12) are called root penalty model, the models in (8),(10) are called

linear penalty model.

Comparing root penalty model with the least square model, the latter is equivalent tointroduce slack variables into equality constraints, and adopt penalty items with a square form to

the objective function, which is:

2

1

min f(x) => obj min f(x) + M (13)r

i

j

obj s

Take two-bus system in Fig.4.1 for example to show the theory of determining key

constraints set with root penalty model. WherePmax < Pa , Qmax > Qa + Qb. The resistance of

FIG 4.1. Two bus system


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22

line a-b is 0. Obviously the active power output can not meet the demand of load. The following

two schemes can be used to restore solvability.

Scheme 1: the power injection of node a is

, 1 2 max., and that of node b is p pa b a bp p P P P

1 22 : the power injection of node a is max, and pab a b abScheme p P P P p p

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

1 ( )obj a bM p p , and one in scheme 2 is 2( )

obj a bM p p clearly 2obj

< 1obj

.

Thus the optimal solution inclines to scheme 2, which can obtain approximate optimal solution

with as few adjustment locations as possible.

4.2.5 The computation efficiency of the extended OPF model.

In the above extended OPF model, the introduction of penalty items to the slack variables

with a large positive number may lead to slow computation speed. In this paper, the

improvement below is made to guarantee the solving speed of extended OPF model by interiorpoint method.

Considering the primal and dual step sizes in section 4.2.2 and 4.2.3, the slack variables become

non-zero values only if the corresponding dual multipliers of slack variables reachM. Because M

is a very large number, it would take moreiterations to for the dual multipliers to reach M, thus

the convergent speed is decreased. The solution is to reduce the coefficients of objective function

as below:

1 1

min f(x)min f(x) + M => (14)r r

j j

j j

s sM

Apparently, same optimal solution can be obtained through this approach, although the value of

objective function is reduced to1/ M of original one.


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23

Chapter 5

Test Results

RTS-24 system is used here to calculate the model proposed in this report using improved

primal-dual interior point method, whereM=1105 and per unit values are used. The reference

power is 100MW.

The symbols used in the following tables denote:

(V): Violation of voltage amplitude in certain node.

(P): Active power injection in certain node.

(Q): Reactive power injection in certain node.

(L): Violation of transmitting power in certain line.

5.1 Results of linear penalty model to restore solvability

RTS-24 system is calculated. Twosituations are considered:

(1) Infeasibility due to strict nodal voltage amplitude constraints.

(2) Infeasibility due to the constraint of transmitting power limit in a certain line.

Under the two situations, computations are carried out as below:

(1) Simulations of model with the introduction of slack variables into equality constraints.

(2) Simulations of model with the introduction of slack variables into inequality constraints.

According to Table I , when the original problem is infeasible due to the extremely strict

voltage amplitude constraints, virtual reactive power injection at several nodes can be obtained

for the model of introducing slack variables to equality constraints, so that the optimal solution

can meet the specific nodal voltage amplitude constraints; several upper or lower nodal voltage

amplitude constraints are broadened for the model of introducing slack variables to inequality

constraints, thus some nodal voltage amplitudes in the optimal solution may violate the current

voltage amplitude constraints.


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24

According to Table II , when transmission line power limits lead to the insolvability of original

problem, the virtual active power injection at several nodes can be obtained for the model of

introducing slack variables to equality constraints; the transmitting limit in the very line which

leads to an unsolvable state can be increased to obtain a approximate solution for the model of

introducing slack variables to inequality constraints. When the transmitting limit in a certain line

keeps reducing, the result of adjustments is constant. In Table II, the transmitting limit in line 1-2

is 1.102. It means that the values of minimum transmitting limits that ensure the solvability of

the original problem in current situation can be obtained.

In sum, when the original problem is feasible, the values of slack variables in extended OPF

model are zero, and it can converge to the same optimal solution as the original problem;

Table I


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25

when it is infeasible, non-zero slack variables are provided, as well as the respective adjustment

scheme.

5.2. Results of root penalty model to determine the key constraints set

Based on the analysis in section 4.2.4 , the adoption of root penalty items can reduce the number

of adjusted locations. Take the insolvable case due to voltage amplitude constraints

for 24-bus system for example, the results comparison between least square model and rootpenalty model are discussed to show how the key constraints set is determined .

Table II


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26

It can be seen in Table III and Table IV that: while slacking equality constraints, virtual reactive

power injections at about 13 nodes are injected to restore the OPF solvability for least square

model; the root penalty model requires far fewer numbers of adjustment locations. For example,

when the range of voltage constraints is 0.99~1.01, only reactive power injection at one node is

needed. So the root penalty model easily detects the critical constraints that lead to insolvability

and is in favor of practical operations.

Table III

Table IV


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CONCLUSION

Generally, when the OPF problem is unsolvable, experiences and repeated trials are required to

find the constraints leading to the insolvable case, which can not provide clear signals to

operation staff. In this paper, an extended OPF model is proposed, which is realized by

introducing slack variables into equality and inequality constraints, adding penalty to objective

function and using improved primal-dual interior point method.

The extended OPF model proposed in this paper has several characteristics as follows:

(1) If the original problem is solvable, the extended model can converge to the same solution,

moreover the iteration number does not increase; if the original problem is unsolvable, there

exist non-zero slack variables, and the results not only point out the insolvability of the original

problem, but also provide adjustment scheme.(2) Slack variables in equality constraints is equivalent to an virtual power injection at certain

nodes, and those in equality constraints is equivalent to broaden the specific security criterions

which lead to insolvability of the original problem.The results of slack variables of equality

constraints are more intuitionistic and easier to operate practically.

(3) The model proposed in this paper can solve the above problem through requiring fewer

adjustment locations, especially for the root penalty model, which can reflect the key constraints

more clearly.


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REFERENCES

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

Flow Measure for Unsolvable Cases Based on Interior Point MethodIEEE 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, Method and 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],IEEE Trans. Power Syst., vol. 13, no. 4,

pp. 12111218, Nov. 1998.

[5] W. Rosehart and J. A. Aguado, Alternative optimal power flow formulations, in Proc.

14th Power System Computation Conf., Seville, Spain, 2002, Session 41, Paper 4.

[6] Wenyuan Li, Juan Yu, Yang Wang, Paul Choudhury, and Jun Sun, Method and 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)

[7] Mingbo Liu, S. K. Yso, Ying Cheng, An extended nonlinear primal-dual interior point

algorithm for reactive-power optimization of large-scale power systems with discrete

control variables,IEEE Trans. Power Syst., vol. 17, no. 4, pp. 982~991, Nov. 2002

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An Extended Optimal Power Flow Measure

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