Download fast voltage stabilty index based optimal reactive power

Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
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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Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
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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Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
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:
≤ (iv) Transformer Tap Limit
≤ V!"#$
%& ≤ %& ≤ %& i€'(
Equation (4) is therefore changed to the following generalized objective function:
, 2
Min ) = ) + ∑*+ − , 2 +∑-. / i€NQglim
i€NVlim
(5)
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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Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
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 C G - 4DE +
>F
GQ2
?
≥ 0
(8)
Rearranging (2), we obtain
HFI .F ?
+JF > K LM? NK LF
<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=
HO F .P
+Q F ?
(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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Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
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 { X iG : 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
X iG := xiG, j : 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:
xiG, j = x Lj + R j ( xiU − x Lj ) , j= 1,2……D where x Lj and xUj 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 v iG , for each solution vector as xiG
G
viG = x BEST
+ P ( x mG − x nG ) ,i=1,2,…….Np
Where x
G
BEST
such that{ x
G
m
(11)
G
n
represents the best solution and { x , x } are two arbitrary vectors at generation G
G
BEST
x mG , x nG } 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
u iG i.e.,
v G
u iG =  iG, j
 xi , j
forR j < CR
otherwise
(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.
u G
xiG +1 =  iG
 xi
forf (u iG ) < f ( xiG )
otherwise
(13)
4. Repeat step 3 until the termination condition is satisfied.
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Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
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 ($/puVAR)
100
6000
1000
30,00,000
TABLE III.Limits
Qc
min
- 0.12
Vg
max
0.35
min
0.9
V load
max
1.1
min
0.96
max
1.05
Tg
min
0.96
max
1.05
6.1 Case Study
Three cases [3] have been studied. Case 1,. Case 2 and 3 are the normal load and heavy
loads.The heavy loads
are 1.25% and 1.5%.. The period of the load level is 8760 hours in
normal,1.25% and 1.5 loading [6].The initial generator bus voltages and transformer taps are set to
1.0 pu. The loads are given as,
Case 1: Pload = 2.834 and Qload = 1.262
Case 2: Pload = 3.542 and Qload = 1.577
Case 3 : Pload = 4.251 and Qload = 1.893
TABLE IV..Initial generations and power losses
Pg
Qg
Ploss
Qloss
Case 1
3.008
1.354
0.176
0.323
Case 2
3.840
2.192
0.314
0.854
Case 3
4.721
3.153
0.461
1.498
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Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
TABLE V..Optimal generator bus voltages.
Bus
1
2
5
8
11
13
Case 1
Case 2
Case 3
1.10
1.10
1.10
1.09
1.10
1.10
1.05
1.09
1.08
1.09
1.10
1.09
1.10
1.10
1.09
1.10
1.10
1.09
TABLE VI.Optimal transformer tap settings.
Branch
(6-9)
(6-10)
(4-12)
(27-28)
Case 1
Case 2
Case 3
1.0433
1.0133
1.0131
0.9540
0.9460
0.9534
1.0118
0.9872
0.9737
0.9627
0.9862
0.9712
TABLE VII.Optimalvar source installments.
Bus
26
28
29
30
Case 1
Case 2
Case 3
0
0.0527
0.0876
0
0.030
0.029
0
0.022
0.027
0
0.031
0.047
TABLE VIII.Optimal generations and power losses
Case 1
Case 2
Case 3
Pg
Qg
Ploss
Qloss
2.989
3.808
4.659
1.288
1.867
2.657
0.159
0.266
0.417
0.266
0.652
1.190
TABLE IX.Cost comparison
PCsave%
Case 1
Case 2
Case 3
8.644
12.452
13.311
WCsave($)
7,98,070.94
19,92,758.84
32,98,528.48
fC($)
8.4225*106
1.4408*107
2.2017*107
TABLE X.Comparision Results
Variables
V1
V2
T6-9
T4-12
QC 17
QC 27
PG
QG
Ploss
Qloss
Case-1
EP[6]
DE
1.05
1.044
1.05
0.975
0
0
2.866
0.926
0.052
0.036
1.05
1.044
1.0433
1.031
0
0
2.989
1.288
0.159
0.266
Case-3
EP[6]
DE
1.05
1.022
0.9
0.95
0.0229
0.196
5.901
2.204
0.233
0.436
1.05
1.022
1.013
0.973
0.297
0.297
4.659
2.657
0.417
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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Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
0RST % =
VZY
QXY
9UVWW
/9UVWW
QXY
9UVWW
x 100 %
\
[KST = hdl ( 0,NKK
−
) = + [
(14)
N]\
0,NKK )
Convergence Rate of the DE Algorithm
18.5
--> Transmissionlineloss (inMW)
Loss Value at Iteration
18
17.5
17
16.5
16
0
20
40
60
80
100
120
--> No. of iterations
140
160
180
200
Figure 2. Converence Rate of DE Algorithm without VAR(100 % Loading)
Convergence Rate of the DE Algorithm
29
--> Transmission line loss (in MW)
Loss Value at Iteration
28.5
28
27.5
27
26.5
0
20
40
60
80
100
120
--> No. of iterations
140
160
180
200
Figure 3. Converence Rate of DE Algorithm witht VAR(1.25 % Loading)
Convergence Rate of the DE Algorithm
47
Loss Value at Iteration
--> Transmission line loss (in MW)
46
45
44
43
42
41
40
0
20
40
60
80
100
120
--> No. of iterations
140
160
180
200
Figure 4. Converence Rate of DE Algorithm with VAR(1.5% Loading)
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Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
6
9.1
Convergence Rate of the DE Algorithm
x 10
cost Value at Iteration
9
--> total cost
8.9
8.8
8.7
8.6
8.5
8.4
0
20
40
60
80
100
120
--> No. of iterations
140
160
180
200
Figure 5.The convergenge characteristic of Normal DE(Case-1) in optimal situation in terms of cost
7
2.5
Convergence Rate of the DE Algorithm
x 10
cost Value at Iteration
2.45
t
2.35
f
c
--> obj unction otal ost
2.4
2.3
2.25
2.2
2.15
0
20
40
60
80
100
120
--> No. of iterations
140
160
180
200
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
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Electrical and Electronics Engineering: An International Journal (ELELIJ) Vol 3, No 1, February 2014
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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