FAST VOLTAGE STABILTY INDEX BASED OPTIMAL REACTIVE POWER PLANNING USING DIFFERENTIAL EVOLUTION

Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014

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FAST VOLTAGE STABILTY INDEX BASED

OPTIMAL REACTIVE POWER PLANNING USING

DIFFERENTIAL EVOLUTION

K.R.Vadivelu1 and Dr.G.V.Marutheswar2

1 Research Scholar,Department of EEE, S.V. University, Tirupati,,India

2 Professor,Department of EEE, S.V. University,Tirupati, India

ABSTRACT

This Article presents an application of Fast Voltage Stability Index (FVSI) to Optimal Reactive Power

Planning (RPP) using Differential Evolution(DE). FVSI is used to identify the weak buses for the Reactive

Power Planning problem which involves process of experimental by voltage stability analysis based on the

load variation. The peak at Fast Voltage Stabilty Index secure to 1 indicates the greatest feasible connected

load and the bus with least connected load is identified as the weakest bus at the point of bifurcation. This

technique is tested on the IEEE 30-bus system. The outcome confirm significant decrease in system losses

and enhancementt of voltage stability with the use of Fast Voltage Stability Index based optimal Reactive

Power Planning using Differential Evolution and compared with Evalutionary Programming

KEYWORDS

Power Systems, Optimal Reactive Power Planning, Fast Voltage Stabilityindex, Differential Evolution.

1.INTRODUCTION One of the most challenging issues in power system research, Reactive Power Planning (RPP).Reactive power planning could be formulated with different objective functions[6] such as cost based objectives considering system operating conditions.Objecives can be changeable and unchanginh Reactive power(VAr)installation cost,real power loss cost and maximizing voltage stability margin.The objective function of the Reactive power planning means to minimize the real power loss and fixed VAr installation cost and deals different constraints are security and voltage stability constraints [1-7]. This different constraints are the key of various classification of optimization models[1-4]. Recently new methods[7] on artificial intelligence have been used in reactive power planning.Conventional optimization methods are based on successive linearization[13] and use the I and II order derivatives of goal function. Since the formulae of RPP problem are hyper quadric functions, linear and quadratic treatment induce lots of restricted minima.The rapid development of power electronics technology provides exciting opportunities to develop new power system equipment for better utilization of existing systems. This Article Presents an application of Fast Voltage Stabilty Index(FVSI) to identify the weak buses for the RPP problem using soft computing technique based Differential Evolution(DE) [15]. Differential Evolution is a method that optimizes a problem by iteratively trying to improve the candidate solution by most versatile implementation maintains a pair of vector population [8, 10, 11,17]. Here, computationally fast indicator of voltage stability index is presented which can be made as direct adjustment to load flow studies called as Fast Voltage Stability Index FVSI [9, 15,16]. The method of determining Fast voltage stability index has been used in the Reactive


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Power Planning problems for the test system and it consists of 6 PV buses, 21 PQ buses and 41 branches.The under[3] load tap setting four branches are (6-9), (6-10), (4-12) and (27-28). The VAr source installation buses are 30, 26, 29 and 25 which are identified based on the proposed technique.

2. NOMENCLATURE List of Symbols Nl = total no of load level durations Nc = total no of possible Reactive Power source installment bus Ni = total no of f buses adjacent to bus i including bus i NPQ = no of of PQ - bus numbers Ng = no of generator bus numbers NT = no of tap - setting transformer branches NB = no of total buses h= energy cost(p.u) dl= duration of load level 1 gk= conductance of branch k Vi= voltage magnitude at bus i θij= voltage angle difference between bus I and bus j ei= fixed VAR source installment cost at busi Cci= per unit VAR source purchase cost at busi Qci= VAR source installed at bus i Qi= reactive power injected into network at busi Gij= mutual conductance between bus i and bus j Bij= mutual susceptance between bus i and bus j Gii,Bii= self conductance and susceptance of bus i Qgi= reactive power generation at bus i Tk= tap setting of transformer branch k NVlim= set of numbers of buses in which voltage over limits NQglim= set of numbers of buses in which VAr over limits

3. PROBLEM FORMULATION It is aimed in this objective function in Reactive Power planning that minimizing of the real power loss (Ploss) in transmission lines of a power system. This is mathematically stated as follows. WC= h ∑ dl ploss,l (1)

Where, (Ploss) , denotes the network real power loss during the period of load level 1. It can be can be expressed in the following equation in the[6] duration dl:

Ploss = ∑�� + �� − 2�� (2)

The second term represents the cost of Reactive Power source installments which has two components, namely, fixed installment cost and purchase cost:

��= ∑�� + ��|��|� (3)

The goal meaning can be expressed as follows:


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Min ��= IC + WC, (4)

Subjected to The functions should satisfy the real and reactive power constraints (equality constraints) (i) Load Flow Constraints:

0 = Pi –Vi ∑ Vj (GijCosθij + BijSinθij)

0 = Qi –Vi ∑ Vj (GijSinθij - BijCosθij)

And also satisfy the inequality constraints like reactive power generation,bus voltage and transformer tap setting limit as follows (ii) Generator Reactive Power Capabilty Limit:

�� ≤ �� ≤ ��

(iii) Voltage Constraints:

�� ≤ �� ≤ V!"#$

(iv) Transformer Tap Limit

%&��≤ %&≤ %&�� i€'(

Equation (4) is therefore changed to the following generalized objective function:

Min )� = )�+ ∑*+�� − ��,��2 +∑-.��/��,��2 (5)

i€NVlim i€NQglim

Subjected to

0 = 0� - �� ∑��1�� + 2�sin�� i € '6/, j€',

0 = �� - �� ∑ ��1��78�� − 2�� i € '9.

j€',

Where,αvi and βQgi are the penalty factors which can be increased in the optimization

procedure;��,�� and ��,��are defined in the following equations:

��,�� ={��7�� < ��

�� 7�� > ��

��,�� ={��7��<��

��if��>��(6)


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4.FVSI FORMULATION The Fast Voltage Stability Index is resulting from the voltage quadratic equation at the receiving bus on a two-bus system [12,14,15]. The general 2-bus representation is illustrated in Figure 1.

Figure 1.Model of Two bus system

From the figure, the voltage quadratic equation at the receiving bus is written as

��-=>? sinδ + cosδB �C��+DE + >F? GQ2 = 0 (7)

set the equation of discriminator be larger than or equal to zero yields

D=>? sinδ+ cosδB �CG�- 4DE + >F

? GQ2 ≥ 0 (8)

Rearranging (2), we obtain

HIF.F?+JF�> K�� LM?�NK L�F

< 1 (9)

since“i”as the sending bus and “j” as the receiving end bus,Since δ is normally very small, then, δ≈0, R Sinδ ≈0 and X receiving bus, Fast Voltage Stabilty Index (FVSI) can be calculated

FVSIij=HOF.P+QF?

(10)

Where ,Z,X are the Impedance and reactance of the line.Where as Qj ,V are the Reactive power at the receiving end and the sending end voltage.

4.1. Procedure For Determining The Maximum Loadability For Weak Bus

Identification Using Fvsi 1. Using Newton-Raphson method ,Run the load flow program for the base case. 2. Estimate Fast voltage stability Index value for all line in the system. 3. Progresively increase the Qj at chosen load bus until the load flow fails to give the results.Calculate Fast Voltage Stabilty Index Values for every load variation. 4. Plot the graph of FVSI versus Q. 5. Take out the line index that has the highest value be s the most critical line with respect to a bus. 6. Select another load bus repeat steps up to 5.


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7. Obtain the voltage at the maximum computable FVSI prior to the divergence of the Load flow .It can be obtained from step 3.This determines the critical Voltage of a Particular bus. 8. Take out the maximum Qj loading for the maximum calculable FVSI for all test bus. It can be obtained from 5.The greatest VAr loading is referred to as the most loadability of a Particular bus. 9. Sort the greatest loadability obtained from step 8 in ascending order and the least loadability maximum is ranked the utmost imply the Weakest bus in the system. 10.Select the feeble buses as the reactive power installation site for the Reactive Power Planning .

5.DIFFERENTIAL EVOLUTION (DE) Differential Evolution is first proposed over 1994-1996 by Storn and Price at Berkeley. Differential evolution (DE) is a population based and parallel search algorithm that operate on the

populations of the possible solution vectors { G

iX : i=1,2,3………,Np} at each generation G

[8,10.11]. Each individual element of the solution vector is composed of D-parameters, namely G

iX := G

jix , : j= 1,2…… D. Various steps in DE are mutation, crossover and selection. The outline

of the DE algorithm is as follows: 1. Initialize the population:

G

jix , = xL

j + )( L

j

U

ij xxR − , j= 1,2……D where L

jx and U

jx are the lower And upper bounds of

the parameter j respectively, and Rj is a random number ,uniformly distributed between [0,1]. 2. Evaluate the population using an objective function. 3. Generate a new population where each new vector is created according to:

(a) Generate a trial vector G

iv , for each solution vector as G

ix

)( G

n

G

m

G

BEST

G

i xxPxv −+= ,i=1,2,…….Np (11)

Where G

BESTx represents the best solution and {G

n

G

m xx , } are two arbitrary vectors at generation G

such that{ G

BESTx G

n

G

m xx , } are mutually different. The constant P is a mutation factor.

(b) Crossover the trial vector and the current vector with crossover probability CR to deliver a baby vector

G

iu i.e.,

<

=

otherwisex

CRforRvu

G

ji

j

G

jiG

i

,

, (12)

(c) Evaluate the baby vector. (d) Use the baby in the new generation if it is at least as good as the current vector; otherwise, the old vector is retained.

(13)

4. Repeat step 3 until the termination condition is satisfied.

<

=+

otherwisex

xfuforfux

G

i

G

i

G

i

G

iG

i

)()(1


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6. NUMERICAL RESULTS The buses for possible VAR source installation is identified using FVSI and the buses are 25, 26, 29 and 30. The greatest loadability and FVSI values for the IEEE 30 bus system are given in Table I.

TABLE I.Bus Ranking and FVSI Values

Rank Bus No Qmax(p.u) FVSI

1 30 0.29 1.014

2 26 0.34 1.042

3 29 0.38 1.043

4 25 0.6 1.020

5 27 0.74 1.071

6 15 0.76 1.007

7 24 0.79 1.014

8 10 0.84 1.003

9 14 0.88 1.012

10 18 0.91 1.018

The parameters and variable limits are listed in Tables II and III. All power and voltage quantities are p.u. value and the base power is used to work out the energy cost.

TABLE II.Parameters.

SB (MVA) h ($/puWh) ei($) Cci ($/puVAR)

100 6000 1000 30,00,000

TABLE III.Limits

Qc Vg V load Tg

min max min max min max min max

- 0.12 0.35 0.9 1.1 0.96 1.05 0.96 1.05

6.1 Case Study Three cases [3] have been studied. Case 1,. Case 2 and 3 are the normal load and heavy loads.The heavy loads are 1.25% and 1.5%.. The period of the load level is 8760 hours in normal,1.25% and 1.5 loading [6].The initial generator bus voltages and transformer taps are set to 1.0 pu. The loads are given as,

Case 1: Pload = 2.834 and Qload = 1.262 Case 2: Pload = 3.542 and Qload = 1.577 Case 3 : Pload = 4.251 and Qload = 1.893

TABLE IV..Initial generations and power losses

Pg Qg Ploss Qloss

Case 1 3.008 1.354 0.176 0.323

Case 2 3.840 2.192 0.314 0.854

Case 3 4.721 3.153 0.461 1.498


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TABLE V..Optimal generator bus voltages.

Bus 1 2 5 8 11 13 Case 1 1.10 1.09 1.05 1.09 1.10 1.10

Case 2 1.10 1.10 1.09 1.10 1.10 1.10

Case 3 1.10 1.10 1.08 1.09 1.09 1.09

TABLE VI.Optimal transformer tap settings.

Branch (6-9) (6-10) (4-12) (27-28) Case 1 1.0433 0.9540 1.0118 0.9627

Case 2 1.0133 0.9460 0.9872 0.9862

Case 3 1.0131 0.9534 0.9737 0.9712

TABLE VII.Optimalvar source installments.

Bus 26 28 29 30 Case 1 0 0 0 0

Case 2 0.0527 0.030 0.022 0.031

Case 3 0.0876 0.029 0.027 0.047

TABLE VIII.Optimal generations and power losses

Pg Qg Ploss Qloss Case 1 2.989 1.288 0.159 0.266

Case 2 3.808 1.867 0.266 0.652

Case 3 4.659 2.657 0.417 1.190

TABLE IX.Cost comparison

PCsave% WCsave($) fC($)

Case 1 8.644 7,98,070.94 8.4225*106

Case 2 12.452 19,92,758.84 1.4408*107

Case 3 13.311 32,98,528.48 2.2017*107

TABLE X.Comparision Results

Variables Case-1 Case-3 EP[6] DE EP[6] DE

V1 1.05 1.05 1.05 1.05

V2 1.044 1.044 1.022 1.022

T6-9 1.05 1.0433 0.9 1.013

T4-12 0.975 1.031 0.95 0.973

QC 17 0 0 0.0229 0.297

QC 27 0 0 0.196 0.297

PG 2.866 2.989 5.901 4.659

QG 0.926 1.288 2.204 2.657

Ploss 0.052 0.159 0.233 0.417

Qloss 0.036 0.266 0.436 1.190

As shown in Table X,Similar results were obtained both approaches for the normal and 1.5% loading.Differential Evolution algorithm has adjusted the voltage magnitude of all load buses and transformer tap settings such that total losses decreased.

7. OPTIMAL RESULTS AND COMPARISION The optimal generator bus voltages, transformer tap settings, VAR source installments, generations , power losses and comparision of cost saving in three cases are obtained as in tables 5 – 9.The real power savings, annual cost savings and the total costs are calculated as,


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0�R�ST% = 9UVWWQXY /9UVWW

VZY

9UVWWQXY x 100 % (14)

[�K�ST = hdl ( 0,NKK��\ − 0,NKKN]\

)

)�= �� + [�

Figure 2. Converence Rate of DE Algorithm without VAR(100 % Loading)

Figure 3. Converence Rate of DE Algorithm witht VAR(1.25 % Loading)

Figure 4. Converence Rate of DE Algorithm with VAR(1.5% Loading)

0 20 40 60 80 100 120 140 160 180 20016

16.5

17

17.5

18

18.5

--> No. of iterations

--> T

ransm

ission line loss (in M

W)

Convergence Rate of the DE Algorithm

Loss Value at Iteration

0 20 40 60 80 100 120 140 160 180 20026.5

27

27.5

28

28.5

29


--> T

ransm

issio

n lin

e loss (in

MW

)



0 20 40 60 80 100 120 140 160 180 20040

41

42

43

44

45

46

47


--> T

ransm

issio

n lin

e loss (in

MW

)




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Figure 5.The convergenge characteristic of Normal DE(Case-1) in optimal situation in terms of cost

Figure 6.The convergenge characteristic of DE(Case-3) in optimal situation in terms of cost

Figures 2, 3 and 4.show the Convergence Rate of Differential Evolution Algorithm with VAR.For normal,1.25% and 1.5 % loading.It can be seen that Convergenge Rate of Differential Evolution Algorithm is capable to arrive at the close vicinity of final solution with in the respective iteration [17]Comparing Differential Evolution Algorithm. Figure 5 and 6..show the convergence rate of Differential Evolution for optimal situation in terms of cost for normal and 1.25% loading.

8.CONCLUSION FVSI based approach has been developed for solving the weak bus oriented RPP problem. Based on FVSI, the locations of reactive power devices for voltage control are determined. The individual greatest loadability obtained from the load buses will be sorted in rising order with the least value being ranked uppermost The highest rank implies the weakest bus in the system with low sustainable load. These are the possible locations for reactive power devices to maintain

0 20 40 60 80 100 120 140 160 180 2008.4

8.5

8.6

8.7

8.8

8.9

9

9.1x 10

6


--> tota

l cost


cost Value at Iteration

0 20 40 60 80 100 120 140 160 180 2002.15

2.2

2.25

2.3

2.35

2.4

2.45

2.5x 10

7


--> o

bj fu

nction

tota

l cost


cost Value at Iteration


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stability of the system. The application studies on the IEEE 30-bus system shows that the optimum Differential Evolution approach gives more savings on real power, annual and the total costs for different loading conditions after transformer tappings are modeled as discrete variables.DE was capable for solving the RPP problem successfully for the case studies,providing a considerable reduction in system losses and an improvement on the voltage profile over the system.The proposed approach a proper planning can be done according to the bus capacity to avoid voltage collapse of the system

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[9] D.E.Goldberg.Gentic Algorithm in Search,optimization and machine learning,AditionWesley,New York,1989

[10] Jani Ronkkonen , SakuKukkonen, Kenneth V. Price, 2005, “Real-Parameter Optimization with Differential Evolution”, IEEE Proceedings, pp. 506-513

[11] H. Yahia, N. Liouane, R. Dhifaoui, 2010, “Weighted Differential Evolution Based PWM Optimization for Single Phase Voltage Source Inverter”, International Review of Electrical Engineering, Vol. 5, Issue. 5, pp. 1956 –1962.

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[14] M. Moghavemmi and F.M. Omar, 1998, “Technique for contingency monitoring and voltage collapse prediction,” IEE Proc. Generation,Transmission and Distribution, vol. 145, pp. 634-640.

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FAST VOLTAGE STABILTY INDEX BASED OPTIMAL REACTIVE POWER PLANNING USING DIFFERENTIAL EVOLUTION

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