Optimal setting of reactive compensation devices with an improved voltage stability index for voltage stability enhancement

Electrical Power and Energy Systems 37 (2012) 50–57

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Optimal setting of reactive compensation devices with an improved voltage stability index for voltage stability enhancement Chien-Feng Yang a, Gordon G. Lai a,⇑, Chia-Hau Lee a, Ching-Tzong Su b, Gary W. Chang a a b

Department of Electrical Engineering, National Chung Cheng University, Chia-Yi 621, Taiwan Jimmy Architects, Chia-Yi 600, Taiwan

a r t i c l e

i n f o

Article history: Received 19 May 2011 Received in revised form 29 October 2011 Accepted 5 December 2011 Available online 9 January 2012 Keywords: Hybrid differential evolution (HDE) Static var compensator (SVC) On-load tap changing (OLTC) transformers Improved voltage stability index

a b s t r a c t Voltage stability improvement is an important issue in power system planning and operation. Voltage stability of a system depends on the network topology and settings of reactive compensation devices. This research first proposes an improved voltage stability index (IVSI) for network systems, and then presents an optimization method for reactive compensation devices settings. To solve the optimization problem, this work introduces the hybrid differential evolution (HDE) to determine tap settings of on-load tap changing (OLTC) transformers, excitation settings of generators or synchronous condensers (SCs), and locations with sizes of static var compensators (SVCs). The performance of this technique is verified using the IEEE 30-bus power flow test system. The results show that the proposed method using the improved voltage stability index and the optimization method is able to effectively enhance voltage stability of a system and reduce line losses. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The voltage stability problem has become a major concern in power systems, especially for a system with heavier loading conditions and without sufficient transmission or generation enhancements. Literature survey shows that a lot of works have been done on the voltage stability analysis of transmission systems. Liu and Vu [1] reconstructed the voltage collapse phenomenon by applying a nonlinear dynamic model of the OLTC, impedance loads, and decoupled reactive power–voltage relations. Popovic et al. [2] studied the dynamic behavior of loads and tap changers for the time of voltage collapse in power systems. Zhu et al. [3] discoursed the effects of OLTC operation on voltage collapse from the conception of how the limit of power transfer from the generation to the load center can be influenced by OLTC operation. Vournas and Karystianos [4] examined the effect of load tap changers on the voltage stability of a power system, and especially for emergency conditions. Chebbo et al. [5] applied the concept of maximum power transfer between two buses to investigate voltage collapse at load buses of the network. The stability index is the ratio of Thevenin’s impedance to load impedance and is at a maximum of 1.0. Chakravorty and Das [6] showed that the load flow solution for radial distribution networks is unique, and presented a new voltage stability index for radial distribution networks. Moghavvemi and Omar [7] presented an approach for calculating ⇑ Corresponding author. Tel.: +886 52720862; fax: +886 52720861. E-mail address: [email protected] (G.G. Lai). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.12.003

voltage stability factor based on a concept of power flow through a single line, and derived the voltage stability factors Lmn to examine system stability. If Lmn exceeds the value of 1.00, the system will lose its stability and thus fall into contingency. He et al. [8] proposed a new approach of locating weak buses in a large scale power system. This new approach is derived using the relationship between voltage stability and the angle difference between sending and receiving end buses. Moghavvemi and Faruque [9] proposed a method to evaluate a static stability indicator for each line of the system; this method has the ability to identify the exact location of voltage collapse in a system. Gu and Wan [10] presented a linearized local voltage stability index based on wide-area measurement systems. Additionally, a lot of algorithms have been presented to solve the optimal reactive power dispatch, Bansilal et al. [11] proposed an expert system for alleviating voltage violations applying switchable shunt reactive compensation and transformer tap settings. Chang and Huang [12] proposed a hybrid optimization scheme applying parallel simulated annealing and a Lagrange multiplier for optimal SVC planning to enhance voltage stability. Thukaram et al. [13] examined effects of the OLTC transformers on voltage stability and recognized critical OLTCs to avoid possibilities of voltage instability conditions. El-Keib and Ma [14] presented an examination on the application of ANNs in voltage stability assessment. Devaraj et al. [15,16] presented an improved genetic algorithm (GE) approach for voltage stability enhancement. Ramirez et al. [17] presented an optimal reactive power dispatch strategy to minimize active power losses and improve voltage stability. Shaheen

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et al. [18] presented a new approach based on DE technique to find out the optimal placement and parameter setting of unified power flow controller (UPFC) for enhancing power system security. Khazali and Kalantar [19] presented a harmony search algorithm for optimal reactive power dispatch. In this work, an improved voltage stability index developed using a network system is proposed. It is suitable application to both radial and network systems. Moreover, the previous researches have not simultaneously determined the settings of OLTCs, generators or SCs, and SVCs. In this paper, a method being able to effectively enhance voltage stability of a system as well as reduce the line losses is proposed. The problem under study is an optimization problem, and the HDE [20–22] is employed to determine tap settings of OLTCs, excitation settings of generators or SCs and locations as well as sizes of SVCs on transmission systems for adjusting reactive power flow and getting optimal voltage stability of a system. The migrant and accelerating operations embedded in HDE are used to overcome traps of local optimal solutions and problems of time consumption.

where Q2 = Q21 is the reactive line power at the receiving end and directing into that end. To obtain a real value for |V2|, the following formula needs to be satisfied.

Pi þ jQ i ¼ V i  Ii

ð7Þ

2. Voltage stability index for the proposed method

n n X X Pi  jQ i ¼ Vi yij  yij V j  Vi j¼0 j¼1

ð8Þ

Several methods have been proposed to assess the static security of power systems, the formulas used to examine the system stability and thus voltage collapse are described below.

½V 1 sinðh  dÞ2  4XQ 2 P 0

ð4Þ

Then a voltage stability index for a line is defined as follows:

Lmn ¼

4XQ 2 ½V 1 sinðh  dÞ2

6 1:0

ð5Þ

2.3. The proposed improved voltage stability index Considering a network shown in Fig. 2, it is easy to develop the power flow formulas, which can be expressed as:

Ii ¼ V i

n X

yij 

j¼0

n X

yij V j

ð6Þ

j¼1

Pi  jQ i ¼ jV i j2

n X

jyij j\hij  jV i j

n X

j¼0

2.1. Voltage stability index by Chakravorty

ð1Þ

It can be further expressed as another form of voltage stability index (VSI) with (2).

VSI ¼

4Z 2 ðP221 þ Q 221 Þ ð2RP 21 þ 2XQ 21 þ jV 1 j2 Þ2

6 1:0

ð9Þ

Separating the real and reactive power in (9) gives

Consider the single line of a radial distribution system shown in Fig. 1, which is part of a radial distribution system. Chakravorty and Das derived a voltage stability index (SI) [6], assuming that the voltage at the sending end is jV 1 j ¼ jV 1 j\d1 and at the receiving end is jV 2 j ¼ jV 2 j\d2 , letting the power angle d = d1–d2, the impedance of the transmission line Z ¼ R þ jX ¼ jZj\h. This index can be expressed as follows:

SI ¼ ð2RP 21 þ 2XQ 21 þ jV 1 j2 Þ2  4jZj2 S221 P 0

jyij jjV j j\ðhij  dij Þ

j¼1

ð2Þ

2.2. Line voltage stability index proposed by Moghavvemi

Pi ¼ jV i j2

n X

jyij j cos hij  jV i j

j¼0

Q i ¼ jV i j2

n X

n X

jyij jjV j j cosðhij  dij Þ

ð10Þ

j¼1

jyij j sin hij  jV i j

j¼0

n X

jyij jjV j j sinðhij  dij Þ

ð11Þ

j¼1

From (10) and (11), we get n n X X ðGij  Bij ÞjV i j2  jV j j½Gij ðcos dij þ sin dij Þ j¼0

j¼1

 Bij ðcos dij  sin dij ÞjV i j  ðPi þ Q i Þ ¼ 0

ð12Þ

From (12), since |Vi| is a real number, the following formula needs to be satisfied:

SI ¼

" n X

#2 jV j j½Gij ðcos dij þ sin dij Þ  Bij ðcos dij  sin dij Þ

j¼1

The voltage stability criterion is based on power transmission concept in a single line [7]. Each single line of a network is first to be assessed before assessing the overall system stability. A single line in a system also illustrated in Fig. 1. The voltage equation can be derived as:

V2 ¼

V 1 sinðh  dÞ  f½V 1 sinðh  dÞ2  4XQ 2 g0:5 2 sin h

Fig. 1. An one-line diagram for developing voltage stability index.

þ4

n X ðGij  Bij ÞðPi þ Q i Þ P 0

ð13Þ

j¼0

The improved voltage stability index (IVSI) we proposed is given below.

ð3Þ

Fig. 2. A network system for developing improved voltage stability index.

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4

IVSI ¼ h Pn

Pn

j¼0 ðGij

j¼1 jV j j½Gij ðcos dij

 Bij ÞðPi þ Q i Þ

þ sin dij Þ  Bij ðcos dij  sin dij Þ

i2 6 1:0 ð14Þ

For an N-bus system, the total number of voltage stability indexes IVSIs is N for all buses of the system. The index IVSI is used in the same way as indexes VSI and Lmn. As the IVSI index for every bus in the system is close to 0, the system is stable. But as long as the IVSI index for any bus in the system is close to 1.0, the system is unstable and voltage collapse may occur. Moreover, the proposed IVSI index is suitable application to both radial and network systems. 2.4. Total voltage stability index Another useful index, total voltage stability index (IVSIT) is proposed in this work. It is used to assess the overall stability for a system with compensation devices, and is used as the objective function for optimizing the settings of compensation devices. For an N-bus system, the IVSIT is the summation of the N voltage stability indexes for all buses of the system. Therefore, the total voltage stability index to the system can be evaluated as follows

IVSIT ¼

N X

IVSIi

obviously observed that bus 3 is the weakest bus of the system for its voltage drop in heavier loading conditions is most significant among the load buses. As SLoad reaches 140%, the voltage at bus 3 will drop to 0.69 p.u., and once SLoad exceeds 140%, the voltage at bus 3 will collapse and lead to system blackout. This critical load value is the voltage stability limit (VSL) of the system. Fig. 5 shows the variations of the IVSI, Lmn and VSI indexes with various loads at bus 3. As SLoad varies from 50% to 140% of the initial load, the indexes IVSI, Lmn and VSI are obviously influenced by the SLoad variation. The larger the SLoad becomes, the more unstable the system will be. Moreover, as SLoad reaches VSL, the IVSI index at bus 3 reaches 1.0 while the Lmn is only 0.45 and VSI index is 0.56, regardless of the system’s blackout. It shows that the Lmn and VSI indexes are not so applicable as IVSI index in examining the voltage stability while the network system is on the brink of blackout due to delivering large load power. This verifies that the IVSI index is really superior to the Lmn and VSI indexes for application to network systems. That is, the IVSI index is suitable for predicting voltage stability index of every bus in any network system, but the Lmn and VSI indexes are suitable for predicting the ones in radial distribution systems. 3. Mathematical model for optimal setting of compensation devices

ð15Þ

i¼1

The voltage deviation index of a bus is defined as the absolute value of the deviation of the bus voltage from one per unit, as shown in (16). The total voltage deviation index of an N-bus system is the sum of the N voltage deviation indexes for all buses, as given in (17)

One problem of this work is to determine simultaneously tap settings of OLTC transformers, excitation settings of generators or SCs, and both locations with sizes of the SVCs for adjusting reactive power flow on transmission systems. This problem belongs to the optimization problem with an objective function to be minimized subject to some constraints. The constraints considered include the limits of bus voltages, OLTC tap settings, capacities of the compensation devices as well as generators and SCs. The mathematical model of the problem can be expressed below:

VDIj ¼ j1  V j j

ð16Þ

Min F ¼ minðIVSIT þ kV  SCV Þ

ð17Þ

where kV is the penalty constant to SCV, and SCV is the sum of the absolute value of the violations to voltage constraints. The optimization problem is subject to operation constraints shown below.

2.5. Voltage deviation index

VDIT ¼

N X

j1  V j j

j¼1

3.1. Limits of voltage stability index for each Bus

2.6. Performance test for the voltage stability indexes To verify the advantage of the proposed index IVSI compared with indexes Lmn and VSI, a six-bus network system shown in Fig. 3 is investigated. It is assumed that the voltage at generator bus is V1 = 1.05\0° p.u., the load at each load bus, SLoad, varies between 50% and 140% of its initial load. Fig. 4 shows the voltage variation at load buses with various load values of the system, it is

6

1:1.025

V2 =1.1

5

0.723+j 1.050

70

09

0.

0.123+j 0.518

07

.4

G1

2

+j 0

Swing

j 0.133

0.008+j 0.370 V1=1.05

ð19Þ

0.282 +j0.640

j 0.300

1

IVSIi < 1 i ¼ 1; 2; 3; . . . ; N

P2 =50MW

G2

30MW +j18Mvar

55MW +j15Mvar

ð18Þ

4

3 1:1.1 55MW +j13Mvar

Fig. 3. 6-Bus power flow test system.

Fig. 4. Voltage variation at load buses with various loads for the 6-bus system.

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C.-F. Yang et al. / Electrical Power and Energy Systems 37 (2012) 50–57

Q kG;min 6 Q kG 6 Q kG;max

for the kth generator and SCs

ð26Þ

where QG,min and QG,max are the reactive power lower limit and upper limit of generators or SCs, respectively. 4. Hybrid differential evolution This paper employs the HDE technique [20–22] to solve the optimization problem of determining tap settings of OLTC transformers, excitation settings of generators or SCs, and locations with sizes of the SVCs. The algorithms of HDE are briefly described below. Step 1. Initialization

Fig. 5. Variations of the three voltage stability indexes with various loads at bus 3 for the 6-bus system.

where IVSIi is the improved voltage stability index of bus i and N is the total number of buses. This constraint can guarantee that during the process of optimizing the objective function (18), all buses satisfy voltage stability index limit. 3.2. Limits of voltage magnitude at each Bus

V min 6 V i 6 V max

i ¼ 1; 2; 3; . . . ; N

ð20Þ

The first step of HDE is to randomly generate an initial population with Np individuals. Each individual consists of Nc control variables and is chosen randomly and would attempt to cover the entire parameter space uniformly. Uniform probability distribution for all random variables is assumed, that is

X 0i ¼ roundðX min þ qi ðX max  X min ÞÞ i ¼ 1; . . . ; Np

ð27Þ

where both Xmin and Xmax are 1  Nc matrixes representing the minimum and maximum values of the Nc control variables, respectively, qi e [0, 1] is a random number, and round () represents the nearest integer of the argument of the round operation. As a result of initialization, the initial population X0 will be an Np  Nc matrix as below

2

X 011

X 012

...

X 022

...

X 01Nc

3

7 7 X 02Nc 7 7 7   7 7 7   7 5 . . . X 0NP Nc

where Vi is the voltage magnitude of bus i, Vmin and Vmax are minimum and maximum bus voltage limits, respectively. N is the total number of buses. The SCV can be obtained from (21) and (22). When Vi satisfies the voltage limitation, Si,CV is zero; otherwise, Si,CV is the difference between Vi and Vmin (if Vi < Vmin) or between Vi and Vmax (if Vi > Vmax).

6 6 0 6 X 21 6 6 0 X ¼6  6 6 6  4 X 0NP1

Si;CV

ð21Þ

The control items represented by variables in each individual are described in detail shown below:

ð22Þ

h X 0i ¼ X 0i1 X 0i2 ... X 0it X 0itþ1 X 0itþ2 ... X 0itþg X 0itþgþ1 i X 0itþgþ2 ...X 0itþgþs X 0itþgþsþ1 X 0itþgþsþ2 ...X 0itþgþsþL

8 if V min 6 V i 6 V max > < 0; ¼ V min  V i ; if V i < V min i ¼ 1; 2; 3; . . . ; N > : V i  V max ; if V i > V max

SCV ¼

N X

Si;CV

i¼1

3.3. Limits of SVCs capacity

Q SVC;min 6 Q SVC 6 Q SVC;max

for each SVC

ð23Þ

where QSVC,min and QSVC,max are the lower limit and upper limit of SVCs, respectively.

for the ith OLTC; i ¼ 1; 2; . . . ; I

ð24Þ

where Tapmin and Tapmax are the lower and upper tap setting limits of OLTCs, respectively. 3.5. Limits of active and reactive power capacity of generators or SCs

PkG;min 6 PkG 6 P kG;max

for the kth generator; k ¼ 1; 2; . . . ; K

 X 0NP2

ð28Þ

ð29Þ

X 0i In (29), denotes individual i which can be any individual of the initial population X0. It has Nc control variables. The variables Xi1  Xit denote the tap settings of OLTCs, X i tþ1  X i tþg denote the excitation settings of generators or SCs, X i tþgþ1  X i tþgþs denote the sizes of the SVCs and X i tþgþsþ1  X i tþgþsþL denote the locations of SVCs. The number of control variables Nc is the sum of t, g, s and L. That is, Nc = t + g + s + L. For example:

X 0i ¼ ½ 0:9 0:9 0:9 0:9 1 1 1 1 1 25 25 25 5 6 7 

3.4. Limits of tap setting of OLTC transformers

Tapimin 6 Tapi 6 Tapimax



ð25Þ

where PG,min and PG,max are the active power lower limit and upper limit of generators, respectively.

In which the variables of individual i include 4 OLTCs, with all their tap settings at 0.9 p.u.; five generators or SCs, with all their voltages at 1 p.u.; three SVCs, with all their reactive power of 25 Mvar; and the locations for placing SVCs at buses 5, 6 and 7. Step 2. Mutation operation A mutant vector is generated based on the present individual X Gi as follows:

Y iGþ1 ¼ roundðX Gi þ MrððX Gr1  X Gr2 Þ þ ðX Gr3  X Gr4 ÞÞÞ

ð30Þ

The mutation rate Mr is selected as Mr e [0, 1.2], and the upper limit of 1.2 for Mr is determined empirically; subscripts i, r1, r2, r3 and r4 e {1, 2, . . . , Np} and r1, r2, r3 and r4 are randomly selected.

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method is then applied to push the present best individual toward a better point. Thus, the acceleration operation can be expressed as:

Step 3. Crossover operation In order to increase the diversity among the individuals of the next generation, a perturbed individual, Y Gþ1 ; and a present indii vidual, X Gi , are chosen by a binominal distribution to progress the crossover operation to generate an offspring. Each gene (variable) of individual i is reproduced from the mutant vector Gþ1 G ðY Gþ1 Þ ¼ ðY Gþ1    Y Gþ1 i i1 Y i2 iNc Þ and the present individual i ðX i Þ ¼ G G G ðX i1 X i2 . . . X iNc Þ. That is:

(

Y iGþ1

¼

X Gi ;

if a random number > C r

Y Gþ1 ; i

otherwise

ð31Þ

where i = 1, 2, . . . , Np; Nc is the dimension of decision parameters and it is the number of genes. The crossover factor is set to Cr e [0, 1], which is determined empirically. Step 4. Estimation and selection The parent is replaced by its offspring if the fitness of the offspring is better than that of the parent. Contrarily, the parent is retained in the next generation if the fitness of the offspring is worse than that of the parent. Two forms are presented as follows:

ðX iGþ1 Þ ¼ arg minfFðX Gi Þ; FðY Gþ1 Þg i

ð32Þ

ðX bGþ1 Þ ¼ arg minfFðX iGþ1 Þg

ð33Þ

where arg min means the argument of the minimum; X Gþ1 is the b best individual, and F is the objective function shown in (18). Step 5. Migration if necessary In order to effectively enhance investigation to the search space and reduce the choice pressure to a small population, a migration operation is introduced to regenerate a new diverse population of individuals. The new populations are yielded based on the best individual X Gþ1 . The j-th gene of individual i is as follows: b

X ijGþ1

8 X Gþ1 X jmin Gþ1 bj < roundðX Gþ1 þ q ðX if q2 < X jmax jmin  X bj ÞÞ; 1 bj X jmin ¼ : Gþ1 roundðX Gþ1 þ q ðX  X ÞÞ; otherwise jmax 1 bj bj

ð34Þ where q1 and q2 are randomly generated numbers uniformly distributed in the range of [0, 1]; i = 1, . . . , Np; j = 1, . . . , Nc. The migration in HDE is executed only if a measure fails to match the desired tolerance of population diversity. This measure is defined as follows:

q¼

NP X Nc X i¼1 i–b

vij =Nc ðNP  1Þ < e1

ð35Þ

j¼1

( b Gþ1 ¼ X b

X bGþ1 ; X bGþ1

if MðX bGþ1 Þ < MðX Gb Þ

 arM 

ðX Gþ1 Þ; b

otherwise

The gradient of the objective function rMðX Gþ1 Þ can be calculated b approximately with finite difference. The step size a e [0, 1] is determined according to the descent property. First, a is set to unity. The b Gþ1 Þ is then compared with MðX Gþ1 Þ. If the objective function Mð X b b descent property is achieved, X Gþ1 becomes a candidate in the next b generation, and is added into this population to replace the worst individual. On the other hand, if the descent requirement fails, the step size is reduced, for example, 0.5 or 0.7. The descent search b Gþ1 called X N at the method is repeated to find the optimal X b b (G + 1)th generation. This result shows the objective function MðX Nb Þ should be at least equal to or smaller than MðX Gþ1 Þ. b Step 7. Repeat Steps 2–6 until the maximum iteration number or the desired fitness is reached 5. Application example The performance of this technique is tested using the IEEE 30bus power flow test system as shown in Fig. 6. This system consists of 30 buses, 41 lines, 2 generators at buses 1 and 2, 4 SCs at buses 5, 8, 11, and 13, 4 OLTC transformers on line 11 (from bus 6 to bus 9), line 12 (from bus 6 to bus 10), line 15 (from bus 4 to bus 12), and line 36 (from bus 28 to bus 27), 3 SVCs and 24 loads. The capacity of each SVC is ±50 Mvar. The line and load data for the test system are given in [23]. The objective function using the total improved stability index IVSIT is shown in (18). Three cases investigated are described below: Case 1: With the excitation settings of generators or SCs being regulable, yet no compensation, increase load of the system from the initial value till causing voltage collapse to find out the voltage stability limit (VSL) and the weakest three buses of the system. Case 2: Enhance voltage stability of the system with the load equaling VSL in Case 1 by optimizing the four compensation conditions shown below. Condition 1: Tap settings of OLTCs and excitation settings of generators or SCs are regulable. All sizes of the SVCs are assumed to be zero and the control variable limits are as below:

where

vij

 Gþ1 Gþ1  8 < 1; if X ij X bj  > e 2   X Gþ1 ¼ bj : 0; otherwise

ð36Þ

Parameters e1 e [0, 1] and e2 e [0, 1] respectively express the desired tolerance for the population diversity and the gene diversity with respect to the best individual. Here Xij is defined as an index of gene diversity. A value of zero for Xij denotes that the j-th gene of individual i is very close to the j-th gene of the best individual. From (35) and (36), it can be seen that the value of q is in the range of [0, 1]. If q is smaller than e1, then the HDE performs the migration to generate a new population to escape the local point; otherwise, the HDE breaks off the migration and keeps an ordinary search direction. Step 6. Accelerated operation if necessary When the fitness in the present generation is not improved any longer using the mutation and crossover operations, a descent

ð37Þ

Fig. 6. IEEE 30-bus power flow test system.

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X i;min ¼ ½ 0:9 0:9 0:9 0:9 0:95 0:95 0:95 0:95 0:95 0 0 0 0 0 0  X i;max ¼ ½ 1:1 1:1 1:1 1:1 1:05 1:05 1:05 1:05 1:05 0 0 0 0 0 0  Condition 2: Excitation settings of generators or SCs and sizes of the SVCs are regulable. The SVCs are located at buses 5, 26 and 30, which are the weakest buses of the system from Case 1; all the tap settings of OLTCs are assumed to be 1.0 p.u. and the control variable limits are as below:

X i;min ¼ ½ 1 1 1 1 0:95 0:95 0:95 0:95 0:95 0 0 0 5 26 30 

Table 1 The values corresponding to the five critical buses for Case 1. Load (%)

Bus Bus 5

Bus 11

Bus 13

Bus 26

Bus 30

100 110 120 130 140 145 148 149

0.1799 0.2295 0.2634 0.3094 0.3879 0.4725 0.5219 0.5493

0.1831 0.1994 0.2241 0.2508 0.2853 0.3562 0.4578 0.4923

0.1188 0.1305 0.1515 0.1726 0.2040 0.2684 0.3144 0.3230

0.082 0.099 0.1192 0.1493 0.1999 0.2818 0.3953 0.5712

0.1160 0.1411 0.1683 0.2107 0.2828 0.3998 0.5786 1.0

X i;max ¼ ½ 1 1 1 1 1:05 1:05 1:05 1:05 1:05 50 50 50 5 26 30  Condition 3: Tap settings of OLTCs, excitation settings of generators or SCs, and sizes of the SVCs are regulable. The SVCs are located at the weakest buses 5, 26 and 30, and the control variable limits are as below: X i;min ¼ ½ 0:9 0:9 0:9 0:9 0:95 0:95 0:95 0:95 0:95 0 0 0 5 26 30  X i;max ¼ ½ 1:1 1:1 1:1 1:1 1:05 1:05 1:05 1:05 1:05 50 50 50 5 26 30 

Condition 4: All the compensation devices (Tap settings of OLTCs, excitation settings of generators or SCs, both locations and sizes of the SVCs) are regulable. The control variable limits are as below: X i;min ¼ ½ 0:9 0:9 0:9 0:9 0:95 0:95 0:95 0:95 0:95 0 0 0 1 1 1  X i;max ¼ ½ 1:1 1:1 1:1 1:1 1:05 1:05 1:05 1:05 1:05 50 50 50 30 30 30 

Case 3: Find out the weakest bus of the system and compare the VSL of the system in Case 1 with that in Case2. The range of control variable limits is from Xi,min to Xi,max. For OLTCs, the tap setting ranges from 0.9 p.u. to 1.1 p.u and each setting interval is 0.0125 p.u. For generators or SCs, the excitation setting ranges from 0.95 p.u. to 1.05 p.u., while the generator at bus 1 is not included in the control variables because bus 1 is a swing bus, and its voltage is assumed to be 1.06 p.u. As for the SVCs, their sizes range from 0 Mvar to 50 Mvar, and their locations can be at any bus of the system. By using the HDE evaluation, the control variables can get their optimal solutions. 6. Simulation results and discussions 6.1. Result of Case 1 Table 1 shows the IVSI at five critical buses of the system with various load values. It is obviously observed that bus 5, bus 26, and bus 30 are the three weakest buses of the system. Fig. 7 shows the IVSI at the three weakest buses of the system with various load values. It is also obviously observed that the load of the system exceeds 149% of the initial load, the IVSI at bus 30 exceeds 1.0 first and then causes voltage collapse. It also indicates that bus 30 is the weakest bus of the system and the VSL of the system is 149% of the initial load. 6.2. Result of Case 2 Table 2 shows the simulation results with 149% of the initial load using HDE. Comparing these four compensation conditions, the superiority of performance on IVSIT in order is Condition 4, Condition 3, Condition 2, and then Condition 1. As to the total power loss (PTLoss), it is Condition 4, Condition 3, Condition 2 and then Condition 1. Finally, as to the minimum voltage (Vsmin) at load buses of the system which is regarded as improvement of the

Fig. 7. The IVSI at the three weakest buses of the system with various load values for Case 1.

lowest voltage at load buses, it is also Condition 4, Condition 3, Condition 2 and then Condition 1. Table 3 shows the IVSI values corresponding to the five weakest buses of the system with 149% of the initial load, it is observed that initially bus 5, bus 26, and bus 30 are the three weakest buses of the system, and their IVSI values can be regarded as the voltage stability indexes of the whole system. In addition, the IVSI value has been reduced appreciably in all the four compensation conditions and Condition 4 is the best one on IVSI enhancement. Fig. 8 depicts the bus voltage profiles at load buses of the system, it can be seen that Condition 4 provides the best voltage profile (0.95 < Vi < 1.05) among those 4 compensations. 6.3. Result of Case 3 As the load increases, the IVSI in each compensation condition increases as well, but the rising rate of IVSI is different from each other. In Table 4, it compares the VSL of the system for Case 3. The VSL of the original system without compensation is 149% of the initial load and bus 30 is the weakest bus. As for the four compensation conditions, they are 157% of the initial load in Condition 1 and bus 5 the weakest bus, 177% of the initial load in Condition 2 and bus 26 the weakest bus, 181% of the initial load in Condition 3 and bus 5 the weakest bus, and even up to 189% of the initial load in Condition 4 and bus 5 the weakest bus. As the main issue considered in this work is the voltage stability limit of a system, the power flow limits of lines are ignored to avoid the problem to be too complex. From the above results analysis, it is observed that the superiority of performance in voltage stability enhancement is Condition 4, Condition 3, Condition 2 and Condition 1 in order. Condition 4 is

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Table 2 Simulation results for Case 2 with 149% of the initial load.

IVSIT VDIT (%) PTLoss (MW) Vsmin (p.u.) Vsmax (p.u.) T6–9 (p.u.) T6–10 (p.u.) T4–12 (p.u.) T28–27 (p.u.) VG2 (p.u.) VC5 (p.u.), Qsc

(Mvar)

VC8 (p.u.), Qsc

(Mvar)

VC11 (p.u.), Qsc

(Mvar)

VC13 (p.u.), Qsc

(Mvar)

Location, QSVC (Mvar), Vsvc (p.u.) Location, Qsvc (Mvar), Vsvc (p.u.) Location, Qsvc (Mvar), Vsvc (p.u.) QSVC, total(Mvar)

Condition 1

Condition 2

Condition 3

Condition 4

2.2692 574.2 68.0 0.7296 0.8606 0.9 0.9 0.9 0.9 1.0456 0.9677, 38.3 0.9717, 38.3 0.9500, 22.5 1.0017, 23.2 0

1.7341 189.5 54.7 0.9027 1.173 1 1 1 1 1.0238 0.9997, 39.6 0.9740, 39.5 1.0258, 23.4 1.0500, 22.7 5, 49.9, 0.9797 26, 40.1, 1.1677 30, 44.6, 1.1728 134.6

1.3062 139.8 51.5 0.9195 1.1307 0.9 0.9 0.925 0.9375 1.0096 0.9587, 37.6 1.0067, 37.8 1.0282, 14.1 1.0500, 23.1 5, 46.9, 0.9487 26, 14.3, 1.0567 30, 37.1, 1.1302 98.3

1.1698 67.9 47.3 0.9510 1.0228 0.9875 0.9125 0.9125 0.9000 1.0417 0.9500, 33.1 1.0379, 37.7 0.9741, 7.5

0 0 0

1.0197, 10.2 7, 50.0, 0.9511 21, 46.6, 1.0250 24, 30.2, 1.0257 126.8

Table 3 The IVSI values corresponding to the five critical buses for Case 2.

Fig. 9. Comparison of convergence characteristics between HDE and GA for Case 2.

and the lowest index VDIT, actually it has the best performances in all respects. To further verify the efficiency of the computed results from HDE, the same optimal problems are also solved by genetic algorithms (GAs) [15]. Fig. 9 shows comparison of computation efficiency and solution quality between HDE and GA. The HDE and GA converge after taking 54 and 59 generations, respectively. It indicates that HDE is faster than GA and the value of objective function obtained by HDE is superior.

Bus

No. compensation

Condition 1

Condition 2

Condition 3

Condition 4

7. Conclusion

5 11 13 26 30

0.5493 0.4923 0.3230 0.5712 1.0

0.4620 0.2647 0.1913 0.1978 0.2782

0.2172 0.2101 0.1436 0.3282 0.1471

0.2680 0.1545 0.1251 0.1717 0.0724

0.2130 0.0115 0.0063 0.1079 0.1600

This study first proposes the index of IVSI which can be used satisfactorily for predicting voltage stability index of every bus in network system. This study then presents a method using the HDE to solve the optimization problem of voltage stability enhancement. The proposed technique can obviously enhance the voltage stability of a system and reduce the line losses. Two application systems are employed to verify the proposed method. Application results show that the proposed method has some advantages. First, the proposed IVSI is better than the previous VSI and Lmn indexes in predicting voltage stability for network systems. IVSI can be more effectively than VSI and Lmn in examining the voltage stability while network system is near voltage collapse and thus system blackout. Second, the proposed method can optimally determine tap settings of OLTCs, excitation settings of generators or SCs and both locations with sizes of SVCs to effectively provide reactive power compensation to enhance the system stability. Thirdly, the migrant and accelerated operations embedded in HDE are used to verify the global optimal solution and consume less time. comparing the employed HDE with the commonly used GA, the HDE can reach better convergence efficiency and objective function value, consequently it increase the ability of the proposed method in enhancing system stability.

1.2

Condition 1 Condition 3

voltage (p.u.)

1.1

Condition 2 Condition 4

1 0.9 0.8 0.7 0.6 0.5

3 4 6 7 9 10 12 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

bus number Fig. 8. Voltage profile at load buses of the system for Case 2.

References

Table 4 Comparison of the VSL and weakest bus of the system for Case 3.

VSL (%) Weakest bus

No. compensation

Condition 1

Condition 2

Condition 3

Condition 4

149 Bus 30

157 Bus 5

177 Bus 26

181 Bus 5

189 Bus 5

the best one, moreover, it not only has the best performance in voltage stability enhancement (i.e., it allows the largest load variation), but also has the greatest power loss reduction (PTLoss)

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