Abstract--In this paper a heuristic technique for solving simultaneous short-term transmission network expansion and reactive power planning problem (TEPRPP) via an AC model is presented. A constructive heuristic algorithm (CHA) aimed to obtaining a significant quality solution for such problem is employed. An interior point method (IPM) is applied to solve TEPRPP as a nonlinear programming (NLP) during the solution steps of the algorithm. For each proposed network topology, an indicator is deployed to identify the weak buses for reactive power sources placement. The objective function of NLP includes the costs of new transmission lines, real power losses as well as reactive power sources. By allocating reactive power sources at load buses, the circuit capacity may increase while the cost of new lines can be decreased. The proposed methodology is tested on Garver´s system and the obtained results shows its capability and the viability of using AC model for solving such non-convex optimization problem.

Index Terms-- Transmission expansion planning, Reactive

power planning, Constructive heuristic algorithm, Non-convex optimization.

I. NOMENCLATURE

0v Investment on new transmission lines

1v Total cost of active power losses and reactive sources c Line costs vector n New lines vector

eK Converted real power to cost

LossP Total real power loss D System operating time QG MVAr size of new reactive power sources vector 1 Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran. 2 Iranian Association of Electrical & Electronics Engineers, Kerman Branch, Kerman, Iran 3 Electrical Engineering Department, Universidade Estadual Paulista, Campus Ilha Solteira. SP, Brazil. *correspounding author:M.Rashidinejad ( mrashidi@ uk.ac.ir ).

u Binary vector that indicates whether or not to install

reactive power sources at load buses. V Voltage magnitude vector of buses θ Phase angle vector of buses

GP Existing real power generation vector

DP Real power demand vector

Q 0G Existing generators reactive power

DQ Reactive power demand vector

GP Min. Limits of generators real power

GP Max. Limits of generators real power

GQ Min .Limits of new reactive sources

QG Max. Limits of new reactive sources 0

GQ Min. Limits of existing generators reactive power

0QG Max. Limits of existing generators reactive power

V Min. Limits of voltage

V Max. Limits of voltage N Diagonal matrices containing vector n

0N Diagonal matrices containing the existing lines fromS Apparent power flow vector "from" the bus toS Apparent power flow vector "to" bus

S Maximum apparent power flow vector

n Maximum number of lines that can be added

ijg Conductance of the transmission lines between ij

ijθ Difference in phase angle between buses i and j

GkQ MVAr size of a reactive source installed at bus k

ku Binary variable that indicates whether or not to

install a reactive source at bus k

0kc Installation cost of reactive source at bus k

1kc Unit cost of reactive source at bus k k Load bus

lΩ Set of all load buses

BN Set of all buses

A.Mahmoudabadi1, 2, M. Rashidinejad1*, M.Mohammadian1, M. Zeinaddini Maymand1, M. Rahmani3, H.Khorasani3

An Application of CHA to Concurrent Short-Term Transmission Expansion & Reactive

Power Planning

Paper accepted for presentation at the 2011 IEEE Trondheim PowerTech

978-1-4244-8417-1/11/$26.00 ©2011

ijb Susceptance of the transmission line or transformer ij shijb Shunt susceptance of the transmission line or

Transformer ij ijn Number of lines between bus i and bus j shib Shunt susceptance at bus i

LV Voltage vector at the load buses

LI Current vector at the load buses

GV Voltage vector at the generator buses

GI Current vector at the generator buses H Hybrid representation of transmission system L Voltage stability limit indicator

II. INTRODUCTION he objective of simultaneous transmission expansion and reactive power planning problem is to determine ‘where’,

‘how many’ and ‘when’ new devices such as: transmission lines, reactive power sources, transformer, among others, must be added to an existing network in order to make its operation viable for a pre-defined horizon of planning at total minimum costs. The benchmark network of the base year, the candidate lines, the power generation and power demand of the planning horizon associated with the investment constraints are the basic data for such problem. TEP has been researched for a long time and a recent review of the literature can be found in [1] while the earlier well cited work is developed by Garver [2]. Heuristic methods [3], meta-heuristic methods [4, 5] and some methods based on conventional optimization [6], can be addressed in literature. Most of the studies referred to above consider only simplified DC models of the network, while recently an accurate AC network modeling has been proposed in [7]. This model has been used in [8] too. In fact, TEP usually divided into three horizons: (1) Long-term (up to 20 years) in which the large transmission interconnections associated with the new energy sources will be considered. (2) Medium-term (up to 10 years) in which more details of the interconnections incorporating alternatives for the regional systems will be determined. (3) Short-term (up to 5 years) where the final adjustments are made regarding to alternatives previously chosen such as reactive compensation while the information of system operation are included to the planning process. Generally in TEP steady-state analysis is usually performed using simplified models system like transportation models, linearized power flow models, etc. The transportation models, the hybrid models, the linear disjunctive model, DC model [9], among others, have been used to achieve the primary topology in first stage. Solving a TEP problem can be handled both by classical optimization techniques [6, 10] as well as meta-heuristics, such as: simulated annealing [11], genetic algorithms [12], tabu search [13] and greedy randomized adaptive search procedure (GRASP) [14]. In a subsequent stage, the expansion plan obtained is checked for other operational constraints. In short-term planning the steady-state

studies uses an AC model in order to assess accurately real power losses and reactive compensation requirements, both for the basic configuration and for contingencies. The use of the complete AC model in the first phase is incipient and there are few technical literatures on the subject. On the other hand, the use of the DC model has the following disadvantages, among others: i) TEP must be separated from the RPP problem. ii) It is frequently necessary to reinforce an expansion plan obtained using DC model considering operational AC conditions. iii) The difficulty of taking into account the power losses in the initial phase of planning. In this paper, an integrated AC TEPRPP introduced, in which it has the following advantages:

1) Efficiently allocating reactive power sources during the planning and consequently decreasing the cost of installation of new lines

2) Using an integrated mathematical model that allows TEP and the optimal allocation of reactive power simultaneously in a unique stage via and AC model.

3) Incorporating the determination of the transmission system precise real power losses in a trivial way.

4) Incorporating nonlinear operational characteristic of new reactive devices such as: FACTS controllers.

5) The possibility of carrying out other types of studies such as: contingency analysis, voltage stability, nodal analysis and transient stability investigation. In this work, a constructive heuristic algorithm (CHA) is used to solve the planning problem using AC model. The advantage of CHA is that it is simple to understand whilst it is robust and very fast. It can be said that CHA still represents a very interesting research area and the solutions derived through such algorithms can be used as a basis to find better solutions using algorithms that impose greater computational costs. One of the difficulties that appear when employing AC model, using CHA, is the necessity to be dealt with nonlinear programming problems. An interior point method (IPM) has been reformulated and adapted to solve such nonlinear problem, as it can be addressed in [15–16]. In this paper, the IPM is used to solve TEPRPP as a NLP problem that should be solved in each step of CHA. In fact, IPM provides a better computational performance for large scale problems than the classical approaches, such as: the Simplex method. In literature, IPM has been used to solve problems like optimal power flow for large-scale systems [17], state estimation [18], load ability maximization [19], voltage stability analysis [20], hydrothermal coordination [21] and security constrained economic dispatch [22]. In this paper for each expansion proposal, an indicator is firstly calculated by performing a normal load flow to identify weak buses for reactive sources allocation. Then, embedding the line proposals and candidate reactive sources into the network an AC OPF is conducted by minimizing the active power losses and the costs of reactive power sources. Employing the results of the AC OPF, constraints deviation associated with line investment, the objective function is evaluated.

T

III. MATHEMATICAL MODEL The mathematical model for TEPRPP problem can be

formulated as following optimization system:

0min Tv c n= (1)

1 e Loss Gminv K P D f (Q ,u )= + (2) s. t.

( , , ) 0G DP V n P Pθ − + = (3)

0( , , ) 0G G DQ V n Q Q Qθ − − + = (4)

G G GP P P≤ ≤ (5)

G G GQ Q Q≤ ≤ (6) 00 0

G GQ Q Q≤ ≤G

(7)

V V V≤ ≤ (8) 0 from 0( ) ( )N N S N N S+ ≤ + (9) 0 to 0( ) ( )N N S N N S+ ≤ + (10)

0 n n≤ ≤ (11) N Integer and θ unbounded

Equations (1) and (2) are two objective functions related to the investment costs of new transmission lines and reactive power sources as well as cost of power losses. Equations (3) and (4) represent the conventional equations of AC power flow considering variable vector n which is the number of circuits (lines and transformers). The limits for real power are represented by Eq. (5), for reactive power of new and existing sources by Eqs. (6) and (7) respectively; and for the voltage magnitudes by Eq. (8). The line flow limits (MVA) are represented by Eqs. (9) and (10). The constraints for the number of new lines are represented by Eq. (11). Lower and upper values for voltage limits are assumed 95% and 105% respectively. Total power loss to be minimized can be defined by Eq. (12).

−+= )]cos(2([ 22

ijjijiijLoss VVVVgP θ (12) The cost of new reactive sources is defined as follows:

0 1( , ) ( )l

G k k Gk kk

f Q u c c Q u∈Ω

= + (13)

The elements of vectors ( , , )P V nθ and ( , , )Q V nθ are calculated by Eqs. (14) and (15), respectively.

( , , ) [ ( )cos ( )sin ]B

i i j ij ij ij ijj N

P V n V V G n B nθ θ θ∈

= + (14)

( , , ) [ ( )sin ( )cos ]B

i i j ij ij ij ijj N

Q V n V V G n B nθ θ θ∈

= −

(15)

The bus admittance matrix elements ( G and B ) are:

0 0

0 0

( ) ( )

( ) ( )l

ij ij ij ij ij

ii ij ij ij ijj

G n n g n gG G n n g n g

∈Ω

= − += = + (16)

0 0

0 0 0

( ) ( )

( ) [ ( )

( ( ) )]l

ij ij ij ij ij

sh shii i ij ij ij

j

shij ij ij

B n n b n b

B B n b n b b

n b b∈Ω

= − +

= = + +

+ +

(17)

Note that in Eqs. (16) and (17), the possibility of a different transmission line or transformer being added in parallel with an existing one in the base case is considered, although the equivalent circuit parameters may be different. It should be noted that off-nominal transformer taps are not considered and in this case both transmission lines and transformers have similar equivalent circuits. This must be mentioned that the present model does not consider the phase shifters. Elements( ij) of vectors fromS and toS in Eqs. (9) and (10) are given by:

2 2( ) ( )from from fromij ij ijS P Q= + (18)

2 ( cos sin )from

ij i ij i j ij ij ij ijP V g VV g bθ θ= − +

(19) 2 ( ) ( sin cos )from sh

ij i ij ij i j ij ij ij ijQ V b b VV g bθ θ= − + − −

(20)

2 2( ) ( )to to toij ij ijS P Q= + (21)

2 ( cos sin )toij j ij i j ij ij ij ijP V g VV g bθ θ= − −

(22)

2 ( ) ( sin cos )to shij j ij ij i j ij ij ij ijQ V b b VV g bθ θ= − + + + (23)

A binary variable vectoru , the connection or disconnection of reactive source to a load bus, are the most important decision variables where any feasible operational solution of power systems depends on their values. The remaining variables only represent the operating state of a feasible solution in which a feasible investment proposal, defined through specified values of n andu , can include several feasible operational states.

IV. WEAK BUS IDENTIFICATION An indicator L is used for identifying weak buses [23].

The indicator uses information of the normal power flow, while the advantage of weak bus identification method lies in the simplicity of the numerical calculation. Since the nature of the calculation is non-iterative then the computational time is decreased. L indicator is a quantitative measure for the estimation of the distance between an actual state of the system and stability limit. The local indicators L facilitate the determination of those nodes from which a collapse may originate. Weak buses can be identified through such index. For calculating L transmission system is represented using a hybrid representation by the following set of equations:

F Y

LL LGL L L

G G GGL GG

ZV I IH

I V VK= = (24)

Where: LLZ , LGF , GLK and GGY are sub-matrices of the hybrid matrix H. The H matrix can be evaluated from the Y bus matrix by a partial inversion, where the voltages at the load buses are exchanged against their currents. This

representation can then be used to define a voltage stability indicator at a load bus, namely jL

which is given by:

01 jj

j

VL

V= + (25)

0 j ji ii GV F V

∈= − (26)

The term jV 0 is a representative of an equivalent generator

comprising the contribution of all generators. The index jL can also be derived and expressed in terms of complex power as the following.

*

2j

jjj j

SLY V

+=+

(27)

j j jcorrS S S+ = + (28)

*

*ji i

jcorr ji Loads ijji j

SZS VVZ∈

≠

= (29)

1jj

jj

Yz+ = (30)

jcorrS Component represents the contributions of the other

loads in power system to the index evaluated at node j.

V. SOLUTION METHODOLOGY Solution methodology proposed in this paper is based on the use of constructive heuristic algorithm (CHA) to perform the decision for network reinforcement by added lines, capacitor banks, and etc. CHA chooses the required lines numbers, capacitor banks via an iterative manner to obtain a good quality solution satisfying the operational constraints. This type of algorithm was used by Garver [2] to find a good-quality solution using the transportation model. CHA may be considered as a step by step search of a good quality solution in a complex multi-objective optimization problem. In the case of TEP problem, in each step one circuit is added to the system (transmission line or transformer).The circuit to be added is chosen using a pre-specified sensitivity index that shows the main characteristic of CHA. The iterative process finishes when a feasible solution generally good quality is found .The solution of the relaxed TEP problem can be used to identify the most adequate circuit that can be added to the existing system. The sensitivity index chosen is the circuit active power flow in the solution of the NLP. To do that, in each step of CHA, the circuit that is to be added to the system is identified by the following sensitivity index: SI = max (nij) (31) Where: nij is the solution of the NLP problem of a relaxed TEPRPP. In each step of CHA, the current topology must be updated. The circuits of the base (initial) topology and the circuits added in the iterative process, form the current topology. One main characteristic of CHA is that the strategy identifies the most important circuit in terms of investment and operational constraints. A simple CHA is proposed here

using the same framework proposed by Garver to solve TEP problem that is explained in the following: i) Assume the base (initial) topology as the current topology. ii) Solve the NLP problem for the current topology. if the solution of the NLP problem indicate that we have a feasible solution with the actual additions, stop (a solution for the TEP problem was found), and then go to step 4. Otherwise continue to step 3; iii) Use the SI to identify the most attractive circuit that can be added to the system. Update the current topology with the addition of the chosen circuit and then go to step 2. iv) Arrange the circuits added in decreasing order costwise. Using NLP problem, verify if it is possible to remove the circuit (and we still have a feasible solution). If yes, remove the circuit, otherwise the circuit must be maintained. Repeat the process, simulating the outage of all circuits. The remaining added circuits represent the solution of CHA. In the CHA shown above, an NLP problem must be solved in steps 2 and 4. The NLP problem is solved using an IPM.

VI. ILLUSTRATIVE TEST The proposed algorithm is implemented in MATLAB platform. For the illustrative tests, four cases were simulated using Garver system. This system has six buses and 15 branch candidates, the total demand is 760 MW, 152 MVAr and a maximum 5 lines can be added to each branch, where the Garver system data is given in [7]. The main objective of the tests is to show that active and reactive power planning can be handled simultaneously. Four different tests have been implemented, while two different base cases have been used. Base case 1 – with the base topology proposed by Garver and Base case 2 – without the base topology proposed by Garver. For each case two tests are managed, TEP without reactive planning and TEPRPP. In test 3 and 4 we consider that

reactive fixed costs 50 10 $c = and reactive variable costs 0.3

$ / VAr and also ek D is set to be 1.

Test-1 Modified Garver system without reactive power allocation considering Base case 1: The algorithm converges after solving 9 NLPs, the planning process resulted in a line investment of US$160000 and the following lines are added: 262 =−n , 253 =−n , 264 =−n Active power losses are 12.117 MW. Fig. 1 shows the complete results of Test1. Test-2 Modified Garver system with reactive power allocation in Base case 1: For this test the final solution is found after solving 7 NLPs and the planning process resulted in a line investment of US$110 million and the following lines are added: 2,1,1 645362 === −−− nnn , Active power losses are 15.373MW and a total 44.09MVAr reactive power source must be installed at bus 2 and bus 5. These buses are weak buses that can be identified using L indicator. Table I shows load buses in ascending order of L indicator. Fig. 2 shows the result of Test2. Test-3 Modified Graver system without reactive power allocation with Base case 2: This test is implemented considering the system has no lines. Only generator and load buses are defined. The algorithm converges after solving 12 NLPs, the planning process resulted in line investment of US$260 million and the following lines are

added: ,3,1,1 623251 === −−− nnn 3,2 6453 == −− nn . Active power losses are 10.987MW. Fig. 3 shows the complete results of the test.

Fig.1. Garver system without reactive power allocation in Base case 1

Fig.2. Modified Garver system with reactive allocation with Base case1

Fig.3. Modified Graver system without reactive allocation with Base case 2

TABLE I - L INDICATOR FOR SYSTEM IN TEST-2 Bus 5 2 4

L Indicator 1.1215 0.4996 0.1785

TABLE II - L INDICATOR FOR SYSTEM IN TEST-4 bus 5 4 2

L Indicator 0.9502 0.5088 0.1755

80.0 MW 160 MVAR

240.00 MW 48.00 MVAR

15

3

2

6 4

160.00 MW 32.00MVAR

40.00 MW 8.00 MVAR

240.00 MW 48.00 MVAR

82.19 MW 36.19 MVAR

80.00 MW 14.24 MVAR

369.89 MW 100.86 MVAR

82.89MW 31.61 MVAR

81.36 MW 16.33 MVAR

78.66 MW 18.54 MVAR

80.01 MW 32.07 MVAR

82.02 MW 14.82 MVAR

77.28 MW 15.34 MVAR

79.34 MW 36.00 MVAR

160.00 MW 48.00 MVAR

243.73 MW 108.37 MVAR

3.502 MVAR

80.67 MW 1.34 MVAR

1.048<0.000

0.95<-9.303

1.0156<0.716

1.05<4.366

0.95<-8.809

0.950<-8.579

26.787MVAR

80.0 MW 160 MVAR

240.00 MW 48.00 MVAR

1 5

3

2

6 4

160.00 MW 32.00MVAR

40.00 MW 8.00 MVAR

240.00 MW 48.00 MVAR

54.24 MW 19.73 MVAR

53.33 MW 10.67 MVAR

304.34 MW 98.38 MVAR

99.92 MW 31.09 MVAR

97.93 MW 11.23 MVAR

78.35 MW 16.42 MVAR

79.66 MW 29.51 MVAR

81.85 MW 26.94 MVAR

47.36 MW 12.26 MVAR

48.09 MW 19.6 MVAR

159.66 MW 45.51 MVAR

80.61 MW 14.49 MVAR

1.05<0.000

0.989<-11.484

1.05<0.283

1.05<-2.803

0.989<-10.461

0.989<-8.492

306.99 MW 117.98MVAR

333.50 MW 101.00 MVAR

1.007<0.000

80.0 MW 160 MVAR

240.00 MW 48.00 MVAR

1 5

3

2

6 4

160.00 MW 32.00MVAR

40.00 MW 8.00 MVAR

240.00 MW 48.00 MVAR

6.19 MW 8.63MVAR

86.82MW 35.87 MVAR

84.42MW 11.86 MVAR

14.87 MW -1.31MVAR

6.13MW 7.96MVAR

97.87 MW 42.05MVAR

95.74MW 20.78 MVAR

25.11MW 13.13MVAR

48.20MW 5.32MVAR

48.69MW 10.19 MVAR

97.82 MW 25.50MVAR

104.59MW 4.47MVAR

108.24MW 40.91MVAR

14.97MW -0.32 MVAR

160.00 MW 47.95 MVAR

281.00MW 112.65MVAR

24.8MW 9.96MVAR

95.90MW 6.34MVAR

0.953<-1.909

1.033<5.296

1.05<12.544

0.95<-5.709

0.982<-5.53

14.095 MVAR

29.993 MVAR

322.22 MW 97.56 MVAR

297.13 MW 128.1 MVAR

240.00 MW 48.00 MVAR

74.48 MW 31.82 MVAR

72.69 MW 13.97 MVAR

1.0<0.000

80.0 MW 160 MVAR

240.00 MW 48.00 MVAR

1 5

3

2

6 4

160.00 MW 32.00MVAR

40.00 MW 8.00 MVAR

14.56 MW 5.88 MVAR

0.2 MW -0.34MVAR

14.41 MW 4.4 MVAR

75.44 MW 24.07 MVAR

74.21 MW 11.78MVAR

21.6 MW 9.14 MVAR

36.29MW 13.26 MVAR

36.61 MW 16.49 MVAR

68.93 MW 21.83 MVAR

72.31 MW 14.47 MVAR

74.09 MW 32.23 MVAR

0.2 MW -0.33 MVAR

21.38 MW 6.94 MVAR

67.90 MW 11.58 MVAR

0.96<-5.011

1.01<3.727

1.05<7.244

0.958<-4.952

0.962<-4.168

152.77 MW 47.51 MVAR

Fig.4. Modified Garver system with reactive allocation with Base case2.

Test-4 Modified Garver system with reactive power allocation in Base case 2:The final test is conducted considering no lines are constructed and no new reactive power sources are installed, just generator and loads are defined, it is assumed that one of the buses can be allocated for reactive power installation. The algorithm converges after solving 13 NLPs with U$190 million line investment and the following lines are added: ,1,2,1 623251 === −−− nnn 253 =−n , 264 =−n . The total active power losses is: 13.561MW. In comparison with test-2, one line is added between buses 2 and 3 and one line is omitted between buses 4 and 6. Two lines are omitted between buses 2 and 6 and with the installation a total 30.3 MVAr at bus 2. It shows that U$50 million in transmission lines is saved. The results of this test are shown in Fig. 4. Table II shows the weak bus indicator for each bus.

VII. ECONOMIC EVALUATION In all the above tests the quality of each solution is improved in comparison with the reported results from literature. In test1 and test 3 the cost of active power losses is decreased while in test 2 and test 4 the amount of reactive sources is also decreased. Table III shows the total cost including line investment, reactive resource cost, the cost of power losses and also the energy saving calculated for each test. By comparing test1 with test 2, and test 3 with test 4 it can be found that when reactive source is used in concurrent with the planning process the total costs will be decreased significantly.

TABLE III - TOTAL INVESTMENT COMPARISON Costs Cost of

Losses (106$)

Line Cost

(106$)

reactive source (106$)

Total Costs (106$)

Cost Saving

%

Test1 [PM]* 12.117 160 …….. 172.117

53 [7]** 12.170 160 …….. 172.170

Test2 [PM]* 15.373 110 13.427 138.800

0.80 [7]** 14.880 110 14.935 139.915

Test3 [PM]* 10.987 260 ……… 270.987

0.152 [7]** 11.400 260 ……… 271.400

Test4 [PM]* 13.561 190 9.290 212.851

7.62 [7]** 12.860 190 26.210 229.070 *proposed method **the method in [7]

VIII. CONCLUSIONS This paper develops a constructive heuristic algorithm for TEPRPP problem based on AC model. The proposed indicator is employed to identify weak buses which may need more reactive compensation. The costs of reactive sources and power losses will be decreased significantly; when mentioned implied. Also, the economic evaluation shows the significant performance of the proposed methodology. It has been shown that proposed method can be improved by adding new indexes to identify weak buses. The economic effectiveness of the obtained results may justify willingness for large scale power systems too.

IX. REFERENCES [1] G. Latorre, R.D. Cruz, J.M. Areiza, and A. Villegas, ‘‘Classification of

publication and models on transmission expansion planning ’’, IEEE Trans. Power Syst, Vol. 18, No. 2, May 2003.

[2] L.L.Garver ‘‘Transmission network estimation using linear programming’’, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-89, No. 8, Sep./Oct. 1970.

[3] A. Monticelli, A. Santos Jr., M.V.F. Pereira, S.H.F. Cunha, B.J. Parker, and J.C.G.Praça , ‘‘Interactive transmission network planning using a least- effort criterion”, IEEE Trans. Power Syst. Vol. PAS-101, No. 10, October 1982.

[4] R.A Gallego, A. Monticelli, and R. Romero, ‘‘Comparative studies on nonconvex optimization methods for transmission network expansion planning’’, IEEE Trans. Power Syst. vol. 13, no. 3, August 1998.

[5] H. Farias Jr., S. Binato, M.G.C. Resende, and D.M. Falcão, ‘‘Power transmission network design by greedy randomized adaptive path relinking’’, IEEE Trans. Power Syst., vol.20, no.1, February 2005.

[6] S. Binato, M.V.F. Pereira, and S. Granville, ‘‘A new benders decomposition approach to solve power transmission network design problems’’, IEEE Trans. Power Syst., vol. 16, no. 2, May 2001.

[7] M.J.Rider, A.V. Garcia and R. Romero, ‘‘Power system transmission network expansion planning using AC model’’, IET Gener. Transm. Distrib, 2007, 1 (5), pp. 731–742.

[8] M. Rahmani , M. Rashidinejad , R. Romero, ‘‘Efficient method for AC transmission network expansion planning ’’, Electric Power System Research, Elsevier, in press 2010.

[9] R. Romero, A. Monticelli, A. Garcia, S. Haffner, ‘‘Test systems and mathematical models for transmission network expansion planning’’, IEE Proc., Gen. Trans. Distrib, 2002, 149(1), pp. 27–36.

[10] R. Romero, A. Monticelli, ‘‘A hierarchical decomposition approach for transmission network expansion planng’’, IEEE Trans. Power Syst., 1994, 9 (1), pp. 373–380.

[11] R.A. Gallego, A.B. Alves, A. Monticelli, R. Romero, ‘‘Parallel simulated annealing applied to long term transmission network expansion planning ’’ IEEE Trans. Power Syst., 1997, 12 (1), pp.181–188.

[12] R.A.Gallego,A. Monteicelli, R.Romero, ‘‘Transmission systems expansion planning by an extended genetic algorithms’’ IEE Proc., Gen. Trans. Distrib., 1998, 145 (3), pp. 329–335.

[13] E.L Da Silva, J.M.A Orteiz, G.C. De Oliveira, S. Binato, ‘‘Transmission network expansion planning under a Tabu search approach’’, IEEE Trans. Power Syst., 2001, 16 (1), pp. 62–68.

[14] Binato, S., De Oliveira, G.C., and De Araujo. ‘‘A greedy randomized adaptive search procedure for transmission expansion planning’’, IEEE Trans. Power Syst., 2001, 16 (2), pp. 247–253.

[15] Torres, G.L., and Quintana, V.H., ‘‘Higher-order interior-point methods for nonlinear OPF solution’’, Proc.32nd North American Power Symp. Waterloo, Canada, 2000, vol. I, no. 6, pp. 45–51

[16] Torres, G.L., and Quintana, V.H. ‘‘on a nonlinear multiple-centrality-corrections interior-point method for optimal power flow’’, IEEE Trans. Power Syst., 2001, 16, (2), pp. 222–228

[17] G.L Torres and V.H. Quintana,‘‘An Interior-point methods for non-linear optimal power flow using voltage rectangular coordinates’’, IEEE Trans. Power Syst., vol. 13, no. 4, pp. 1211-1218, Nov. 1998.

[18] H. Wei, H. Sasaki, J. Kubokawa and R. Yohoyama, ‘‘An interior point methods for power systems weighted nonlinear L1 norm static state estimation’’, IEEE Trans. PWRS, vol. 13, no. 2, pp. 617-623, May. 1998.

[19] G.D. Irisarri, X. Wang, J. Tong and S. Mokhtari, ‘‘Maximum loadability of power systems using interior point method nonlinear optimization,’’ IEEE Trans. Power Syst., vol. 12, no. 1, pp. 162-172, Feb. 1997.

[20] X. Wang, G.C. Ejebe, J. Tong and J.G. Waight, “Preventive/corrective control for voltage stability using direct interior point method”, IEEE Trans. PWRS, vol. 13, no. 3, pp. 878-883, Aug. 1998.

[21] J. Medina, V.H. Quintana, A.J. Conejo and F.P. Thoden, ‘‘A comparison of interior-point codes for medium-term hydro-thermal coordination’’, IEEE Trans. Power Syst, vol. 13, no. 3, pp. 836-843, Aug. 1998.

[22] X. Yan and V.H. Quintana, ‘‘an efficient predictor corrector interior point algorithm for security-constrained economic dispatch’’ IEEE Trans. Power Syst, vol. 12, no. 2, pp. 803-810, May. 1997.

[23] Kessel, H Glavitsch, ‘Estimating the voltage stability of a power system’, IEEE Trans on Power Delivery, vol. PWRD-1, No.3, July 1986, pp. 346-354