Optimal Allocation of TCSC Device Based on Particle Swarm Optimization Algorithm

Optimal Allocation of TCSC Device Based on Particle Swarm Optimization Algorithm

P. Ramesh * and K. Padma**

*P.G. Scholar, **Assistant Professor, Department of Electrical Engineering, A.U.C.E (A),A.U., Visakhapatnam-530003

Abstract

This paper presents an application of Particle Swarm Optimization (PSO) to find out the optimal location and the optimal parameter settings of TCSC devices under single contingency to eliminate the bus voltage violations and improvement of power flow in a transmission line. The particle swarm optimization (PSO) technique is used to solve the optimal power flow problem for steady-state studies by maintaining thermal and voltage constraints. The suitability of the proposed approach is examined on the standard IEEE 30-bus test system with TCSC FACTS device. Simulation results show that proposed PSO algorithm gives better solution to enhance the system performance with TCSC device compared to without TCSC FACTS device.

Keywords: Newton Raphsons Method, TCSC, FACTS device, Particle Swarm Optimization (PSO) technique, Optimal Power Flow, Cost of Generation.

1. Introduction

   The operation mechanism of power system becomes more and more complicated due to the continuously increasing load demand which leads to an augmented stress of the transmission lines and higher risks for faulted lines. Therefore, power system

   can be operated in less secure state following unexpected line congestions and low voltages. Construction of new transmission lines can be one solution for leading more stable and secure operation of power systems. But it becomes a time-consuming process due to political and environmental reasons. Because of all that, it becomes more important to control the power flow along the transmission lines to meet the needs of power transfer [1]-[2].

   Power flow is a function of transmission line impedance, the magnitude of the sending end and receiving end voltages and the phase angle between voltages. By controlling one or a combination of the power flow arrangements, it is possible to control the active as well as the reactive power flow in the transmission line [3].

   A new solution to improve the stability and security of the power system is the Flexible AC Transmission Systems FACTS devices [4] can improve the stability of the power network, reduce the flows of heavily loaded lines by controlling their parameters, and maintain the bus voltages at desired level. Consequently they can improve the power system security in contingency [5].

   With FACTS technology [6]-[7] , such as Thyristor Controlled Series Compensator (TCSC), Static Var

   Compensators (SVCs), Static Synchronous Compensators (STATCOMs), Static Synchronous Series Compensators (SSSCs) and Unified Power Flow Controller (UPFC) etc ,bus voltages, line impedances and phase angles in the power system can be regulated rapidly and flexibly. Thus, FACTS can facilitate the power flow control, enhance the power transfer capability, decrease the generation cost, and improve the security and stability of the power system.

   Thyristor Controlled Series Compensator (TCSC) is one of the most effective FACTS devices which offer smooth and flexible control of the line impedance with much faster response [8]. TCSC can also enhance the transient stability of power system that means if there is any sudden change of load that occurs in power system it will maintain stability without loss of synchronism by using TCSC FACTS device is connected in series with the transmission line and also increase the transfer capability of the transmission system by reducing the transfer reactance between the buses at which the line is connected. However, to achieve the above mentioned benefits, the TCSC should be properly installed in the network with appropriate parameter settings [9]

   Thus in this paper, TCSC FACTS controller is

   incorporated to solve an optimization problem with different objectives such as minimization of cost of generation, real power loss, enhancement of voltage profile and voltage angles and L-index as these are the basis for improved system performance. The particle swarm optimization (PSO) based algorithm is used effectively to solve the optimal power flow problem [10], it present great characteristics and capability of

   determining global optima, incorporating a set of constraints including voltage stability and FACTS device. In order to calculate the power losses and check the system operating constraints such as voltage profile, a load flow model is used. An existing Newton-Rapson load flow algorithm is introduced [11]. This model is further modified to incorporate TCSC FACTS device into the network and PSO technique is applied to the modified model to enhance the performance of the power system. Thus, effectiveness of the proposed method was tested on standard IEEE 30-bus system and comparison was made on the performance of without and with TCSC FACTS device.

   The organization of this paper is as follows. Section

   2 addresses the Computation of Voltage Stability Index (L-index), FACTS controller such as TCSC is explained in Section 3, Mathematical formulation of OPF problem is given in section 4, Overview of Particle Swarm Optimization Algorithm is represented in section 5, Overall Computational Procedure is given in the section 6, the simulation results is given in section 7, and Comparison of fuel cost of generation with different OPF models is illustrated in section 8, and finally the conclusion is given in section 9.

2. Voltage Stability Index (L-index) Computation

   The voltage stability L-index is a good voltage stability indicator with its value change between zero (no load) and one (voltage collapse) [5]-[6]. Moreover, it can be used as a quantitative measure to estimate the voltage stability margin against the operating point. For a given system operating condition, using the load

   flow (state estimation) results, the voltage stability L – index is computed as [5],

   system behavior and enhancement of system reliability. However, their main function is to control power flows.

   L j = 1

   g

   i1

   Vi

   V

   Fji

   j

   (1)

   3.1. Thyristor Control series Compensator (TCSC)

   j g 1,…, n

   All the terms within the sigma on the RHS of equation (1) are complex quantities. The values of F ji are btained from the network Y-bus matrix. For

   stability, the index L j must not be violated (maximum limit=1) for any of the nodes j. Hence, the global

   One of the important FACTS controllers is the TCSC which allows rapid and continuous changes of the transmission line impedance. TCSC [14]-[16] controls the active power transmitted by varying the effective line reactance by connecting a variable reactance in series with line and is shown in Figure 1.

   TCSC is mainly used for improving the active power

   indicator L j

   describing the stability of the complete

   flow across the transmission line.

   subsystem is given by maximum of L j for all j (load

   buses). An

   L j -index value away from 1 and close to

   0 indicates an improved system security. The

   advantage of this

   L j -index lies in the simplicity of the

   numerical calculation and expressiveness of the results.

3. FACTS controllers

   FACTS controllers are able to change, in a fast and effective way, the network parameters in order to

   Figure 1. Circuit diagram of TCSC

   The active and reactive power equations at bus k are:

   k k kk

   achieve better system performance. FACTS controllers such asphase shifter, shunt, or series compensation and

   Pk VkVm Bkm sin(k m )

   (2)

   the most recent developed thyristor controlled based

   power electronic controllers, make it possible to

   Q V 2 B

   VkVm Bkm cos(k m )

   (3)

   control circuit impedance, voltage angle and power flow for optimal operation performance of power systems, facilitate the development of competitive electric energy markets, stimulate the unbundling the power generation from transmission and mandate open access to transmission services, etc. The benefit brought about by FACTS includes improvement of

   For the power equations at bus m, the subscripts k and

   m are exchanged in Equations (2) and (3).

   In Newton-Raphson solutions these equations are linearized with respect to the series reactance. For the condition, where the series reactance regulates the amount of active power flowing from bus k to bus m at

   km

   a value P reg , the set of linearized power flow

   equations is solved to minimize fuel cost of generation maintaining

   P

   Pk

   Pk

   Pk V

   k

   Pk V

   m

   Pk X

   thermal and voltage constraints can be formulated as

   k

   k

   m

   Vk

   Vm

   X TCSC

   TCSC k

   follows [17]-[18]:

   P

   P P

   P V

   P V

   P X

   V

   m m m m

   k

   m m

   m

   TCSC

   m NG

   k

   Q

   m

   Q

   Vk

   Q

   Vm

   Q

   X TCSC

   Q

   k

   Minimize F = ((a P2 b P C )

   i Gi i Gi i

   (6)

   Q

   k k k V k V

   k X V

   k

   V k

   V m X

   TCSC

   k

   i1

   k m k m

   TCSC

   Vm

   m

   Q

   Qm

   Qm

   Qm V

   V k

   Qm V

   V m

   Qm X

   X

   TCSC Vm

   The minimization problem is subjected to

   k m k m

   TCSC

   X

   following equality and inequality constraints

   P XTCSC P XTCSC P XTCSC P XTCSC

   P XTCSC

   TCSC

   P XTCSC km km km V km V

   k

   m

   km X

   X TCSC

   k

   km V V

   m k m

   X TCSC

   TCSC

   (4)

   1. Equality Constraints

      These are the sets of nonlinear power flow

      Where

      TCSC

      P

      km

      is the active power flow mismatch

      equations that govern the power system, i.e.

      n

      PGi PDi Vi V j Yij cos (ij i j ) 0

      (7)

      for the series reactance

      j 1

      n

      P

      P

      TCSC

      km

      reg km

      XTCSC,cal

      • P

      km

      QGi

      • QDi

      • Vi

        j1

        V j Yij

        sin(ij

        i

      • j

        ) 0

        (8)

        X is given as

        Where

        PGi

        and

        QGi are the real and reactive

        TCSC

        power outputs injected at bus i , the load demand at the

        XTCSC

        X

        i TCSC

        i1 TCSC

        same bus is represented by

        PDi and QDi , and elements

        • X

        and is the incremental change in series reactance; and

        X

        of the bus admittance matrix are represented by

        andij .

        Yij

        P TCSC,cal

        is the calculated power.

        km

        The state variable

        X TCSC of the series controller is

   2. Inequality Constraints

   These are the set of constraints that represent the

   updated at the end of each iterative step according to system operational and security limits like the bounds

   X

   (i)

   on the following:

   X

   X

   (i)

   TCSC

   (i1)

   TCSC

   • TCSC

   X

   (i1)

   X

   TCSC

   (5)

   1. Generators real and reactive power outputs

      TCSC

      Pmin P

      Pmax ,i 1,, N

      (9)

      This changing reactance represents the series reactance regulates the amount of active power flowing

      from bus k to bus m at a specified value P reg

      Gi

      Q

      min

      Gi

      Gi

      QGi

      Gi

      Gi

      Qmax ,i 1,, N

      (10)

      km

      i

4. Mathematical formulation of OPF

   1. Voltage magnitudes at each bus in the network

      Problem

      Mathematically, the OPF problem with FACTS is

      min

      V

      i

      Vi

      V max ,i 1,, NL

      (11)

   2. Transformer taps setting PSO is basically developed through simulation of bird

      T

      min

      i

      Ti

      T max ,i 1,, NT

      (12)

      flocking in two-dimension space. The position of each individual (agent) is represented by XY axis position

      i

   3. Reactive power injections due to capacitor banks Modification of the agent position is realized by the

      Qmin Q

      Qmax ,i 1,,CS

      (13)

      position and velocity information.

      Ci Ci Ci

   4. Transmission lines loading

      An optimization technique based on the above concept can be described as follows: namely, bird

      i

      i

      S S max ,i 1,, nl

      (14)

      flocking optimizes a certain objective function. Each agent knows its best value so far (pbest) and its XY

   5. Voltage stability index position. Moreover, each agent knows the best value so

      Lji

      Ljmax ,i 1,, NL

      (15)

      far in the group (gbest) among pbests [20]. Each agent tries to modify its position. This modification can be

      i

   6. TCSC device constraints: Reactance constraint of TCSC

   represented by the concept of velocity. Velocity of each agent can be modified by the following equation:

   X min X

   X max

   (16)

   vk 1 wvk c rand * ( pbest sk ) c rand

   * (gbest sk )

   TCSC

   TCSC

   TCSC

   i i 1 1

   i i 2 2 i

   The equality constraints are satisfied by running the power flow program. The generator bus real power

   generations ( Pgi ), generator terminal voltages ( Vgi ),

   P

   of

   transformer tap settings ( Ti ), the reactive power

   Where,

   v k : Velocity of agent i at iteration k,

   i

   w : weighting factor,

   (17)

   generation of capacitor bank ( QCi ), and

   reg km

   c1 & c2: cognition and social components,

   TCSC are control variables and they are self-restricted by the representation itself. The active power

   generation at the slack bus ( Pgs ), load bus voltages

   (VLi ) and reactive power generation ( Qgi ), line flows ( Si ), and voltage stability ( L j )-index are state

   variables which are restricted through penalty function approach.

5. Overview of Particle Swarm

   rand: random number between 0 and 1

   s

   i

   k : Current position of agent i at iteration k. pbesti : the pbest of agent i.

   gbest : gbest of group.

   i i i

   Using the above equation, a certain velocity which gradually gets close to pbest and gbest can be calculated. The current position (searching point in the solution space) can be modified by the following equation:

   Optimization

   PSO is one of the optimization techniques and belongs to evolutionary computation techniques [19].

   sk 1 sk vk 1

   (18)

6. Overall Computational Procedure for Solving the Problem

   The implementation steps of the proposed PSO based algorithm can be written as follows;

   Step 1: Input the system data for load flow analysis Step 2: Select FACTS devices and its location in the

   sytem

   Step 3: At the generation Gen =0; set the simulation parameters of PSO parameters and randomly initialize k individuals within respective limits and save them in the archive.

   Step 4: For each individual in the archive, run power flow under the selected network contingency to determine load bus voltages, angles, load bus voltage stability indices, generator reactive power outputs and calculate line power flows.

   Step 5: Evaluate the penalty functions

   Step 6: Evaluate the objective function values and the corresponding fitness values for each individual.

   Step 7: Find the generation local best xlocal and global best xglobal and store them.

   Step 8: Increase the generation counter Gen= Gen+1.

   Step 9: Apply the PSO operators to generate new k individuals

   Step 10: For each new individual in the archive, run power flow to determine load bus voltages, angles, load bus voltage stability indices, generator reactive power outputs and calculate line power flows.

   Step 11: Evaluate the penalty functions

   Step 12: Evaluate the objective function values and the corresponding fitness values for each new individual.

   Step 13: Apply the selection operator of PSO and update the individuals.

   Step 14: Update the generation local best xlocal and global best xglobal and store them.

   Step 15: If one of stopping criterion have not been met, repeat steps 4-14. Else go to stop 16

   Step 16: Print the results

7. Simulation Results

   The proposed PSO algorithm is employed to solve optimal power flow problem incorporating TCSC FACTS device for enhancement of system Performance is tested on standard IEEE 30-bus test system.

   The PSO parameters used for the simulation are summarized in Table 1

   Table 1. Optimal parameter settings for PSO

   Parameter	PSO
   Population size	20
   Number of iterations	150
   Cognitive constant, c1	2
   Social constant, c2	2
   Inertia weight, W	0.3-0.95

   The network and load data for this system is taken from [21]. To test the ability of the proposed PSO algorithm one objective function is considered that is minimization of cost of generation. In order to show the affect of power flow control capability of the

   FACTS device in proposed PSO OPF algorithm, two sub case studies are carried out on the standard IEEE 30-bus system.

   Case (a):power system normal operation (without FACTS devices installation),

   Case (b): One TCSC device is installed at an optimal

   line connected between buses 9 and 10 with

   Control Variables	Limits(p.u)		Without FACTS	With FACTS
   	Min	Max		TCSC
   PG1	0.50	2.000	1.7718	1.7748
   PG2	0.20	0.800	0.4867	0.4888
   PG3 PG4	0.10 0.10	0.350 0.300	0.2109 0.1215	0.2080 0.1183
   PG5	0.15	0.500	0.2144	0.2147
   PG6	0.12	0.400	0.1200	0.1202
   VG1	0.95	1.10	1.0868	1.0854
   VG2	0.95	1.10	1.0667	1.0656
   VG3	0.95	1.10	1.0405	1.0382
   VG4 VG5	0.95 0.95	1.10 1.10	1.0645 1.0348	1.0369 1.0338
   VG6	0.95	1.10	1.0425	1.0442
   Tap – 1	0.9	1.1	1.0464	1.0114
   Tap – 2	0.9	1.1	0.9000	1.0187
   Tap – 3	0.9	1.1	0.9568	0.9651
   Tap – 4	0.9	1.1	0.9623	0.9831
   QC10	0.0	0.10	0.0783	0.0993
   QC12 QC15	0.0 0.0	0.10 0.10	0.0000 0.0629	0.0000 0.0264
   QC17	0.0	0.10	0.0518	0.0276
   QC20	0.0	0.10	0.0785	0.0666
   QC21	0.0	0.10	0.0386	0.0693
   QC23 QC24	0.0 0.0	0.10 0.10	0.0429 0.0260	0.0337 0.0415
   QC29	0.0	0.10	0.0260	0.0336
   Cost ($/h)			800.8678	800.5671
   Ploss (p.u.)			0.0913	0.0904
   Ljmax			0.1381	0.1301
   CPU time (s)			45.5510	53.8600

   Table 2. Optimal settings of control variables for IEEE 30- bus system.

   line real power flow Pji

   base case values.

   is 1.25 times of

   The series reactance ( X TCSC ) ratings of TCSC is

   set at 50% of the transmission-line, thus inductive Reactance is 0.055.

   The first case is the normal operation of network without using any FACTS device, in second case optimal location of one TCSC device has been considered.

   From Table 2. it shows that the details of the control variables and the installation of TCSC in the network gives the best performance of the system compared to the without FACTS device in the network in terms of reduction in cost of generation, power loss reduction, maximum of voltage stability indices. It also gives that PSO algorithm is able to enhance the system performance while maintaining all control variables and reactive power outputs within their limits The convergence characteristic of the cost of generation without and with TCSC (one at a time) at

   optimal location is shown in figure 2

   Figure 2. Convergence of cost of generation without and with TCSC FACTS device for IEEE 30-bus system

   The Figures 3-6 shows the percentage MVA loading of the lines, voltage profiles, voltage angles, and voltage stability indices of buses without

   and with TCSC at optimal location

   Figure 3. Percentage MVA line loadings of IEEE 30- bus system after optimization without FACTS and with TCSC FACTS device

   Figure 4. Voltage profiles of IEEE 30-bus system after optimization without and with TCSC FACTS device.

   Figure 5. Voltage angles of IEEE 30-bus system after optimization without and with TCSC FACTS device.

   Figure 6. Voltage stability indices of IEEE 30-bus system after optimization without and with TCSC FACTS device

8. Comparison of fuel cost of generation with different OPF models

   The comparison of fuel cost of the proposed method with those of the methods reported in the literature is given in Table 3. It can be seen that PSO algorithm gives less cost of generation compared with the cst of generation obtained with other OPF methods.

   Method	Fuel Cost ($/hr)
   EP [22]	802.907
   TS [23]	802.502
   TS/SA [23]	802.788
   ITS [24]	804.556
   IEP [25]	802.465
   SADE_ALM [26]	802.404
   OPFPSO [19]	800.410
   MDE-OPF [27]	802.376
   Genetic Algorithm ($/hr) [28]	803.050
   Gradient method [29]	802.430
   PSO (proposed) without FACTS	800.8678
   PSO (proposed) with TCSC	800.5600

   Table 3. Comparison of fuel costs for IEEE 30-bus System

   (TCSC) scheme is very effective compared to without FACTS in improving the security of the power system.

   • TCSC FACTS device can increases the power transfer capability of the transmission line by increasing the voltage at their terminals of both ends of transmission line and also decrease the line reactance for improving power transfer capability.

   • Proper selection of FACTS device and its location can effectively improve the overall system performance.

     From Table 3 It can be seen that the PSO algorithm with TCSC gives less cost of generation compared with the cost of generation obtained with other OPF methods

9. Conclusions

   This paper has presented an OPF model incorporating FACTS controller such TCSC using the PSO algorithm for enhancement of system performance. This model is able to solve power networks of any size and converges with minimum number of iterations and independent of initial conditions. The standard IEEE 30-bus system has been used to demonstrate the proposed method over a wide range of power flow variations in the transmission system. The simulation results shows that proposed OPF with thyristor Controlled Series Compensator

   • Finally, the optimization problem with different objectives such as minimization of cost of generation, reduction in power loss, enhancement of voltage profile and voltage angle at every bus and L-index are improved system performance

References

1. P.M. Anderson and A.A. Fouad. Power SystemControl and Stability, Revised printing, IEEE Power System EngineeringSeries, IEEE Press Inc., 1993.

2. A.L.Bettiol, L.Wehenkel, M. Pavella. Transient Stability Constrained Maximum Allowable Transfer. IEEE Trans. on PWRS,Vol. 14, No 2: 654659,May 1999.

3. R. Rajaraman, F. Alvarado, A. Maniaci, R.Camfield, S. Jalali,Determination of location and amount of series compensation to increase power transfer capability, IEEE Trans. Power System vol. 13, pp. 294 299,1998.

4. N. G. Hingorani and L. Gyugyi, Understanding FACTS concepts and technology of flexible AC transmission systems. New York: IEEE Press, 2000.

5. T.T. Lie, W. Deng, Optimal flexible AC transmission systems (FACTS) devices allocation, Int. J. Electic. Power Energy System Vol. 19 pp. 125134, 1999.

6. IEEE Power Engineering Society/CIGRE, FACTS Overview, IEEE Service Centre, Piscataway, N.J., 1995, Special issue, 95TP108

7. Claudio A. Canizares and Zeno T. Faur, Analysis of SVC and TCSC controllers in voltage collapse, IEEE Trans. on Power Systems, Vol. 14, No.1, pp. 158-165,

   February 1999.

8. C.R.Fuerte-Esquivel, E.Acha, and H.Ambriz- Perez, A thyristor controlled series compensator model for the power flow solution of practical power networks, IEEE Trans. Power System, vol. 15, No. 1, pp.58-64, Feb. 2000.

9. S. Meikandasivam, Rajesh Kumar Nema and Shailendra Kumar Jain,"Behavioral Study of TCSC Device A Matlab/Simulink Implementation,"World Academy of Science, Engineering and Technology, Vol. 45, 2008, pp 694-699.

10. M. Saravanan, S. M. R Slochanal, P. Venkatesh and J. P. S. Abraham,"Application of Particle Swarm Optimization Technique for Optimal Location of FACTS Devices Considering Cost of Installation and System Loadability",Electric Power Systems Research, Vol. 77, No. 3/4, 2007 , pp. 276-283.

11. C.R.Puerle-Esquivel and E. Acha, A Newton-type algorithm for the control of power flow in

    electricalpower networks, IEEE Trans. Power System, vol. 12, No. 4, pp. 1474-1480, Nov. 1997

12. Hsiao-Dong Chiang and Rene Jean Jumeau,Toward a practical performance index fodetecting voltage collapse in electric power systems, IEEE Transactions on power systems, Vol.10, No.21, 1992, pp.584-592.

13. P Kessel and H Glavitsch, Estimating the voltage stability of a power system, IEEE Trans. on PD, Vol.1, No.3, pp.346-354, 1986.

14. A.R.Messia, M.A.Perez, and E.Hernandez,"Co- ordinated application of FACTS devices to enhance study-state voltage stability," International Journal on Electrical Power and Energy Systems, vol. 25, no. 4, pp. 259-267, May 2003.

15. Enrique Acha, Claudio R.Fuerte Esquivel, Hugo Ambriz-Perez, Cesar Angeles- Camacho.FACTS modeling and simulation in power networks .

16. AR. Etemad, H.A. Shayanfar and R. Navabi , Optimal location and setting of TCSC under single line contingency using mixed integer nonlinear programming, International Conference on Environment and Electrical Engineering (EEEIC), pp , 2010, 250-253.

17. Alsac O., Stott B. Optimal load flow with steady state security, IEEE Trans Power Appar Syst 1974; PAS-93; 745-51.

18. P. E. O. Yumbla, J. M. Ramirez, C. A. Coello Coello. Optimal power flow subject to J.security constraints solved with a particle swarm optimizer, IEEE Transactions on Power Systems, vol. 23, no. 1, Feb., 2008.

19. Abido MA. Optimal power flow using particle swarm optimization, Electric Power Energy Syst 2002; 24(7): 563-71

20. Kennedy and R. Eberhart (1995), Particle Swarm Optimization, Proc. of IEEE Conf. on Neural Networks (ICNN), Vol. IV, pp. 1942-1948, Perth, Australia.

21. IEEE 30-bus system (1996), (Online) Available

    at //www.ee.washington.edu

22. Yang H, Yang P, Huang C (1996). Evolutionary programming based economic dispatch for units with non-smooth fuel cost functions.IEEE Trans. Power Syst., 11: 112-118.

23. Ongsakul W., Bhasaputra P., Optimal power flow with FACTS devices by hybrid TS/SA approach, Electr. Power Energ. Syst., 2002, 24, 851-857.

24. Lin W, Cheng F, Tsay M (2002). An improved tabu search for economic dispatch with multiple minima. IEEE Trans. Power Syst., 17: 108-112.

25. W.Ongsakul and T.Tantimaporn, Optimal power flow by improved evolutionary programming, Electric Power Components and Systems, 34:79-95, 2006.

26. Thitithamrongchai, C., Eua-arporn, B., 2007. Self- adaptive differential evolution based optimal power flow for units with non-smooth fuel cost functions. J.

    Electrical Systems 3 (2), 88-99

27. Samir Sayah, Khaled Zehar. Modified differential evolution algorithm for optimal power flow with non-smooth cost function. Energy conversion and Management 2009:49:3036-3042.

28. D Devaraj and B. Yegnanarayana, Genetic Algorithm-Based Optimal Power Flow for Security Enhancement, IEE Proceedings on Generation,

    Transmission and Distribution 2005, 152(6), pp 899-

    905.

29. J. Abadie and J. Carpentier, "Generalization of the Wolfe Reduced Gradient method to the case of non linear constraints", Revue Optimisation, pp 37

-47, Academic Press, 1969.