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Probabilistic assessment and preventive control of voltage security margins using artificial neural network
Electrical Power and Energy Systems 29 (2007) 99–105 www.elsevier.com/locate/ijepes
Probabilistic assessment and preventive control of voltage security margins using artificial neural network L.D. Arya a
a,*
, L.S. Titare b, D.P. Kothari
c
Electrical Engineering Department, Shri. G. S. Institute of Technology and Science, Indore (M.P.) 452003, India b Department of Electrical Engineering, Government Engineering College, Jabalpur, (M.P.) 482011, India c Centre for Energy Studies, I.I.T., Delhi, New Delhi 110016, India Received 18 February 2004; received in revised form 29 March 2006; accepted 15 May 2006
Abstract This paper describes a new technique for probabilistic assessment and preventive control of voltage security margins using artificial neural network. The probabilistic insecurity index (PISI) has been obtained for various operating conditions considering single and double line outages and accounting static voltage stability limit. Evaluated PISI has been obtained for various settings of reactive power control variables and loading conditions using cut-set method. These results have been used to train a multi layer feed forward network so as to assess the security on line. Further sensitivities of PISI have been evaluated based on trained network and have been used to control power system security. The algorithm has been implemented on a 6-bus, 7-line IEEE standard test system. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Voltage stability; Security; Probabilistic insecurity index; ANN
1. Introduction Power system security studies i.e. assessment and control forms an integral part of modern energy management systems (EMS). Power companies are facing a major challenge in maintaining quality and security of power supply due to ever increasing interconnections and loadings in the large power system networks. Economic constraint has forced the utilities to operate their generators and transmission systems to their maximum loadability point. Transient stability (synchronous) limits of the system have increased substantially with the development of improved protective devices, SVS, fast turbine valving and automatic voltage regulators. These developments have allowed to flow more real power over longer distances and systems which, used to be transient stability limited, are now facing the problem of maintaining the required bus voltages and *
Corresponding author. E-mail addresses: ldarya@rediffmail.com (L.D. Arya), lstitare@ yahoo.co.in (L.S. Titare), [email protected] (D.P. Kothari). 0142-0615/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2006.05.005
has become voltage stability limited [1]. Precisely voltage security has been defined as the ability of a system, not only to operate stably, but also remain stable (as far as the maintenance of the system voltage is concerned) following any reasonably credible contingency or adverse system change [2,3]. Further it has been a standard practice to evaluate reliability indices (probabilistic approach) based on security analysis accounting correctability, in planning stage [4–6]. If failure probability or as termed at times probability of load curtailment (PLC) is less, then robustness of the system is more towards violation of operating constraints. Evaluation of PLC requires large number of power flow studies under various loadings as well as outage conditions. Such type of index can not be used to assess security of the system in operating condition. Development of artificial neural network can make such failure probability indices applicable in real time applications in power systems. Most of the research work in evaluating the reliability indices did not account voltage security from voltage stability viewpoint. Billinton and Khan [7] used LP model for adequacy assessment of the composite power
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system accounts voltage and line flow constraints. MonteCarlo sampling with a variance reduction technique was developed by Pereira and Pinto [8] for reliability evaluation of a transmission system. A decomposition-simulation approach was used for power system reliability evaluation by Deng and Singh [9]. Billinton and Li [10] developed a system state transition sampling method for transmission system reliability evaluation. Melo et al. [11] and Billinton and Aboreshaid [12] accounted voltage stability constraints in evaluating the reliability indices. Melo et al. were probably the first to present an approach to calculate voltage collapse related indices as well as their impact on power system adequacy indices based on restoring system solvability by load shedding. Billinton and Aboreshaid developed a new methodology, which accounts voltage stability considerations in adequacy assessment of transmission network. In Ref. [6] a methodology has been developed for evaluating failure probability of a bulk power system based on peak load consideration and accounting correctability of the system. This has been achieved by optimizing static voltage stability limit using univariate search method and voltage stability limit have been obtained using a predictor–corrector based continuation power flow. Capitanescu and Cutsem [17] presented a multi contingency sensitivity based approach for preventive control of voltage security margins utilizing deterministic approach. In this paper a new viewpoint has been presented for voltage security assessment and control using reactive power rescheduling and if required even using load shedding. A probabilistic insecurity index (PISI) has been evaluated using cut-set method by considering single and double line outages. Voltage stability limits have been calculated using continuation power flows in each single and double line outage condition. The PISI have been obtained for various single and double line outages and for various reactive power control variables and loading conditions, using these results a multi layer ANN is trained to obtain PISI for any operating conditions. This trained network is then used to obtain sensitivities of PISI with respect to reactive power control variables and load sheds at various buses, as the input to network is reactive power control variables and load at each buses. For this closed form relations have been derived for sensitivities relating load-shed and reactive power control variables to PISI. These sensitivities are used for voltage securities control either using reactive power control variables and if necessary load shedding at selected buses. Application of ANN makes it suitable for on line applications. Next section describes the methodology for obtaining probabilistic insecurity index.
are directly related to the modes of system failure and therefore identify the distinct and discrete ways in which a system may fail. In identifying cut-set one identifies the most important line or combination of lines, which are of greatest importance. A minimal cut-set is a set of transmission line which, when failed, causes failure of the system but when any one element of the set has not failed, does not cause system failure. All component of a minimal cut-set must be in the failure state to cause system failure. Failure probability of a power network can be written as follows: P F ¼ P ½C 1 þ C 2 þ þ C n ;
where C1,C2,. . .,Cn are minimal cut-sets. This precise evaluation is always theoretically possible, but it is exhaustive and time-consuming exercise, which becomes prohibitive with power network. To overcome this problem, approximations are usually made in the evaluation of failure probability, which although reduce precision, permit much faster evaluation. The degree of inaccuracy introduced is usually negligible and within the tolerance associated with the data of the transmission line availability for power system, which have large values of repair rates and low value of failure rates for each transmission line. Hence, the failure probability is approximated by retaining first order terms in the expansion of Eq. (1) as follows: n X PF ¼ P ðC i Þ: ð2Þ i¼1
The second approximation is to neglect cut-sets of an order greater than a certain value. Number of components comprising that cut-set is equal to the order of a cut-set. In present paper cut-sets up to second order have been considered. This is due to the fact that probability of outage of more than two transmission lines is negligible. Further higher order cut-set under usual loading conditions may not be minimal cut-sets, which are required in Eq. (1). Hence relation (2) is utilized for evaluating probability of failure (PF) under above mentioned approximation. Errors in calculation of unreliability may even less than 1%. A two-state model (up state and down state) is used to model the operation of each transmission line. The up state indicates that the transmission line is in operating state and down state implies that the element is inoperable due to a failure or scheduled off. Fig. 1 shows the two-state model. In this figure ki and li are failure and repair rates of transmission line, respectively. This model is used to provide availability and unavailability of transmission line in long run expression for these are given as follows [14]:
2. Cut-set method and evaluation of probabilistic insecurity index The cut-set method is a powerful one for determining the reliability of a power network. The method can be easily programmed on a digital computer for fast and efficient solution of any general network. Moreover cut-sets
ð1Þ
λi Down state
Up state
μi Fig. 1. Two state model for each transmission line.
L.D. Arya et al. / Electrical Power and Energy Systems 29 (2007) 99–105
Ai ¼ li =ðki þ li Þ
ð3Þ
Ai ¼ ki =ðki þ li Þ
ð4Þ
Ai and Ai denote availability and unavailability of transmission line. Minimal cut-set (first and second order) for a specified load is obtained by comparing the load with static voltage stability limit in base case as well as under various line outage conditions. Static voltage stability limits under single or double line outage condition is obtained using continuation power flow methodology [13]. For adequate system operation sufficient stability margin must be maintained. Using the result of continuation power flow, one can prepare a capacity outage table, which gives permissible loading and corresponding state of the network. Now for various load levels one identifies cut-set from the capacity availability table. Relation (2) is utilized to get probability of failure. This probability of failure is termed as probabilistic insecurity index (PISI) of transmission system. It is further stressed that loadability limit depends on the settings of Perform base case load flow solution and obtain static voltage stability limit
Perform continuation power flow for single and double line outage condition (k) and obtain Plimit for each contingent condition
1
2
101
reactive power control variables. Hence failure probability or PISI is obtained for various settings of reactive power control variables. Computational sequences are shown in Fig. 2 with the help of a flow chart. In block 3–5 static voltage stability limit is obtained for various specified loads and cut-sets are obtained. In block-6 failure probability PF is calculated. In block-7 settings of reactive power control variable is altered in a mono variable fashion to generate PISI for various settings of control variables. 3. Evaluation of sensitivity of probabilistic insecurity index (PISI) with respect to reactive power control variables using artificial neural network Instances obtained in previous section are used to train a multi layer feed forward network. This network contains one input layer, one hidden layer and one output layer. The network is trained using back propagation algorithm [18]. Number of units in input layer equals to number of reactive power control variables and total number of load buses. Number of unit in output layer is one, which gives output as PISI. Further the neurons in the hidden layer are assumed to be sigmoidal. Neuron in output layer is assumed to be non-sigmoidal (linear). Network equations of Fig. 3 are written as follows: m X O¼ W jo Oj ðnon-sigmoidalÞ: ð5Þ j¼1
Output of jth neuron in hidden layer is as follows: Prepare capacity-outage probability table and contingencies indicating P(k) limit
Identify for various peak load consideration cut-sets C1(i), C2(i),...., Cr(i) for ith loading condition i = 1,...., NI
Calculate PF failure probability using eq.(9)
Change the control setting one by one in steps above and below up to limits Up = Up + ΔUp
3
Oj ¼ 1=ð1 þ eNetj Þ; where Netj is n X Netj ¼ W ij X i ;
4
j ¼ 1; 2; . . . ; m;
ð6Þ
i ¼ 1; . . . ; n:
ð7Þ
i¼1
In above equations Wjo are the weights connected between jth hidden neuron and output neuron. Wij are the weight Wij
5
U1
Σ
O1
U2
Σ
O2
W10 W20
6
U3
Σ
W30 O3
Σ
UNC
No
If all change have been made?
O (Output)
Wmo
P1
7 PNB
Σ
Om
Yes Input layer
Stop Fig. 2. Flow chart for computing probabilistic insecurity index.
Hidden layer (‘m’ units)
Output layer
Fig. 3. Generalised diagram of back propagation network for voltage security control and PISI calculations.
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connected between ith input node and jth hidden neuron. Xi is input variable at ith node [19]. Weight change DWjo are given by following formulae:
uled. Correction for such control variable is obtained as follows:
DW jo ¼ g d Oj ; d ¼ ðT OÞ;
where PISIO is the present insecurity index as obtained by the ANN, PISIth is the threshold value of the indicator and SPISr is the sensitivity of indicator with respect to rth control variable. If control correction as obtained using relation (14) violates the limit it can be put to limiting value. Then again modified sensitivities are evaluated and another control variable is selected for control action. This process of monovariable rescheduling terminates when PISI is brought within limits. At last load flow program is executed and it is verified that all voltages are within limits. If it is found that all reactive power control variables have been exhausted and even at this moment PISI violates the limit then sensitivities of PISI are evaluated with respect to load shed at the load buses and a bus is selected for load shed having highest sensitivity as follows:
ð8Þ ð9Þ
T and O are target value and output of network, respectively. ‘g’ is learning rate lies between [0, 1]. Expression for weight change DWij is given as follows using Back propagation algorithm: DW ij ¼ g dj X i ;
ð10Þ
where dj is error gradient and is given as follows: dj ¼ d:W jo :Oj ð1 Oj Þ;
ð11Þ
Xi is the element of input vector [X], where T
T
½X ¼ ½U 1 ; U 2 ; . . . ; U NC ; P 1 ; P 2 ; . . . ; P NB : Once the ANN of Fig. 3 is trained the network Eqs. (5)–(7) are used for calculating sensitivities of ‘O’ (PISI) with respect to input variables either control variables or disturbance variables (loads). Sensitivity expression is derived using chain rule of differentiation as follows: m X oPISI=oX i ¼ ½ðoPISI=oOj ÞðoOj =oNetj ÞðoNetj =oX i Þ j¼1
¼ SPISi ;
ð12Þ
where SPISi is the sensitivity of PISI with respect to Xi. Algebraic manipulation of network equations gives following expression for sensitivities: m X ½W jo Oj ð1 Oj ÞW ij : ð13Þ SPISi ¼ j¼1
Evaluation of sensitivities as governed by Eq. (13) is very simple and follows quickly from Fig. 3. Expression (13) can be introduced as sum of all path gains from an input node to output node. Path gain is the multiplication of all weights and gain of the nodes {Oj(1 Oj)} encountered in the path. This rule of calculating the sensitivity will be even applicable even for many hidden layers. The voltage security control algorithm is explained in next section. This algorithm also includes the possibility of load shedding.
DU r ¼ ðPISIO PISIth Þ=SPISr ;
ð14Þ
DP i ¼ ðPISIO PISIth Þ=SPIDi ;
ð15Þ
where SPIDi is sensitivity of PISI with respect to load shed at ith bus. This should not exceed a maximum amount of load shed at a bus (say 80% of total load at a bus, as 20% load is firm load and may be required for emergency services). Hence, if total load shed at a bus is insufficient, next bus is selected for load shed having highest sensitivity. This procedure continues till PISI is brought within limit. In a sense the ANN act as a system wise relay, which not only detects the insecure condition but also suggests for corrective action. This relay or smart software provides security to the system from voltage stability viewpoint. 5. Results and discussion The algorithm developed for voltage security control using rescheduling of reactive power control variables or load shed has been implemented on a 6-bus, 7-line IEEE standard test system [16]. Fig. 4 represents the test system. The system has two generating buses whose voltages can be adjusted between 0.95 6 Vi 6 1.15 pu for controlling the voltages of load buses. Availabilities of lines from line no.1 to line no. 7 are assumed as 0.99636033, 0.99636033,
1
4
3
2
4. A monovariable approach for voltage security enhancement Trained ANN is used for on-line assessment of voltage security and further sensitivities can be evaluated for corrective rescheduling. A threshold value of PISI can be selected for this purpose. At any instant, if the index PISI crosses the limit, evaluated sensitivities of this index with respect to control variables are used for corrective rescheduling. Control variable with highest sensitivity is resched-
6
5
Fig. 4. Line diagram of 6-bus system.
L.D. Arya et al. / Electrical Power and Energy Systems 29 (2007) 99–105
103
Table 1 Training instances for training ANN S. no
V1 (pu)
V2 (pu)
Pd3 (pu)
Pd4 (pu)
Pd5 (pu)
Pd6 (pu)
PISI as obtained by cut-set method
PISI as obtained by ANN
% Error
1 2 3 4 5 6 7 8
1.00 1.00 1.00 1.05 1.05 1.05 1.10 1.10
1.05 1.05 1.05 1.10 1.10 1.10 1.15 1.15
0.1925 0.4400 0.6050 0.4400 0.6325 0.7425 0.6050 0.8250
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.1050 0.2400 0.3300 0.2400 0.3450 0.4050 0.3300 0.4500
0.1750 0.4000 0.5500 0.4000 0.5750 0.6750 0.5500 0.7500
0.003446762 0.015166344 0.026306507 0.011581222 0.019627254 0.026306507 0.015166344 0.026306507
0.0031868 0.0159991 0.0246776 0.0110873 0.0211902 0.0270174 0.0148240 0.0264666
07.54 05.49 06.19 04.26 07.96 02.70 02.25 00.60
PISI as obtained using cut-set method and ANN.
0.99658703, 0.99636033, 0.99636033, 0.99658703, and 0.99658703. These typical figures have been taken from Ref. [15]. Probability of failure or PISI has been calculated for various settings of PV buses and loading conditions using the methodology of Section 2. Static voltage stability limits have been used for calculating PISI, which were obtained using predictor–corrector technique of Ref. [13]. Table 1 shows the values of PISI as calculated using cutset method and as obtained using trained network. This table also shows the percentage error in its last column. Table 2 gives various weights Wjo and Wij connected between hidden layer and output neuron and those between input layer and hidden layer. Numbers of neurons in hidden layer are six. Variables, put to input layer, are V1, V2, P3, P4, P5, and P6 i.e. voltages at generating buses and loads. Trained ANN has been validated with few testing instances other than used in training. The PISI as obtained using the trained network and cut-set method have been shown in Table 3 along with percentage errors for test cases. There is no rule for proportion between training and testing instances. In the present study training instances have been selected so as to encompass wide range of operating conditions for the power system. Further testing instances have been selected in this range, other than training instances. The trained network has been tested for 50 test instances. Only few (10) are given in Table 3 having highest magnitudes of errors. Values of probability of failure as termed PISI should not exceed a specified value (threshold value) say 0.01851. This value has been selected such that in all outage condition distance of voltage collapse point (in terms of MVA) is at least 20% of static voltage stability limit. This is an adequacy requirement. Depending on the operating philosophy and requirement of individual power utility this may vary. This threshold value is required in Eqs. (14) and (15) for evaluating control actions DUr or DPi. The probability of failure as obtained with trained ANN has percentage error less than 10%. If one want to reduce error than size of the ANN can be increased. This may be achieved by increasing number of neurons in hidden layer and increasing the hidden layers. Corrective rescheduling has been demonstrated for an operating condition
Table 2 Weights as obtained after training the back propagation network of Fig. 3 S. no
Weight between ith input node and jth hidden neuron (Wij)
W11 W12 W13 W14 W15 W16 W21 W22 W23 W24 W25 W26 W31 W32 W33 W34 W35 W36 W41 W42 W43 W44 W45 W46 W51 W52 W53 W54 W55 W56 W61 W62 W63 W64 W65 W66 W11 W21 W31 W41 W51 W61
0.15702530 0.14933560 0.03406646 0.03600000 0.20349430 0.09815425 0.24467800 0.27443130 0.06061339 0.10500000 0.05960826 0.04098819 0.10991660 0.02265394 0.10150800 0.17300000 0.03701656 0.01903466 0.27204240 0.22949380 0.20038590 0.18000000 0.08702040 0.22870280 0.04429901 0.06504500 0.02363667 0.07200000 0.00983370 0.09505729 0.08332255 0.04221593 0.09263467 0.18600000 0.13874920 0.21287110
Weight between jth hidden neuron and Oth output node (Wjo)
0.07430349 0.28554220 0.00087990 0.49452950 0.00048859 0.05809121
V1 = 0.95 pu, V2 = 1.00 pu and total load Pd = 1.7 pu. At this moment stability margin obtained is less than 20%. PISI
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Table 3 Validation of trained network using test cases as generated using cut-set method S. no.
V1 (pu)
V2 (pu)
Pd3 (pu)
Pd4 (pu)
Pd5 (pu)
Pd6 (pu)
PISI as obtained by cut-set method
PISI as obtained by ANN
% Error
1 2 3 4 5 6 7 8 9 10
0.95 0.95 1.00 1.00 1.05 1.05 1.05 1.10 1.00 1.10
1.00 1.00 1.05 1.05 1.10 1.10 1.10 1.15 1.05 1.15
0.1925 0.3850 0.3575 0.5775 0.5225 0.6875 0.4400 0.5500 0.5500 0.7150
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.1050 0.2100 0.1950 0.3150 0.2850 0.3750 0.2400 0.3000 0.3000 0.3900
0.1750 0.3500 0.3250 0.5250 0.4750 0.6250 0.4000 0.5000 0.5000 0.6500
0.00797143 0.018512031 0.011581222 0.022972705 0.015166344 0.022972705 0.011581222 0.011581222 0.022972705 0.019627254
0.0080719 0.0180551 0.0116980 0.0232249 0.0154400 0.0240996 0.0110874 0.0119379 0.0217746 0.0206276
01.25 02.46 01.00 01.09 01.54 04.90 04.26 03.08 05.21 05.09
Table 4 Voltages, loads before and after load shed for 6-bus test system S. no.
1 2 3 4 5 6
Bus voltages in pu
Load at buses in pu
Before load shed
After load shed
Before load shed
After load shed
1.1500 1.1500 0.8921 0.9127 0.8649 0.8834
1.1500 1.1500 0.9272 0.9526 0.9128 0.9446
0.0000 0.0000 0.9325 0.0000 0.5772 0.8251
0.0000 0.0000 0.9325 0.0000 0.5772 0.5171
on-line by training ANN using BPA,(iii) closed form relations have been derived to evaluate the sensitivity of PISI with respect to control variables and load shed and (iv) using the developed sensitivities, an algorithm has been developed for voltage security control and load shedding. The algorithm can be applicable on-line using the trained ANN. Since voltage instability is a phenomenon with slow dynamics [20]. Hence assessment and preventive actions are possible on-line using this ANN. References
obtained at this condition is 0.02238 pu. This is more than the specified threshold value. Sensitivities were evaluated at this moment. Voltage correction at bus No.1 is obtained using the sensitivity PISI with respect to this bus as DV1 = 0.076 pu. This brings PISI within limit (0.01851 pu) as final value. Another case taken at V1 = 1.15 pu, V2 = 1.15 pu at total load 3.30 pu, at this operating point PISI is 0.0283271 pu. It is observed at this point that correctability with reactive power reschedule is not possible. Hence sensitivities were evaluated with respect to load shed at the various load buses. The bus no. 6 having highest sensitivity 0.0319595 have been selected for load shedding. The amount of load shed as calculated using relation (15) is 0.307905 pu is within limit. The PISI calculated as 0.01850 pu, which is less than the threshold value. Voltages, loads before and after load shed are given in Table 4. 6. Conclusions An algorithm with a new viewpoint has been presented for voltage security assessment and control based on probability of failure of a system. The probabilities of failure for various operating conditions have been obtained using cutset method accounting static voltage stability limit. These cases have been used to train ANN using BPA. Further BPA has been used to obtain sensitivity of PISI with respect to control variables and load shed to maintain at least the threshold value of PISI. Main contributions of this paper are (i) development of an algorithm for evaluating PISI using cut-set method accounting static voltage stability limit,(ii) the applicability of PISI has been made
[1] Taylor CW. Power system voltage stability. Mc Graw Hill Inc; 1994. [2] Concordia C. Voltage instability. J Elec Power and Energy Syst 1991;13(1):14–24. [3] IEEE System Dynamic Performance Subcommittee. Voltage stability of power systems concepts, Analytical tools and industry experience. IEEE document 90th 0358-2-PWR-1990. [4] Sebastian Rios M, Victor Vidal P, Kiguel David L. Bus based reliability indices and associated costs in the bulk power system. IEEE Trans Power Syst 1998;13(3):719–23. [5] Billinton R, Wenyuan L. Hybrid Approach for reliability evaluation of composite generation and transmission systems using Monte-Carlo simulation and enumeration technique. IEE Proc C 1991;138(3): 233–41. [6] Arya LD, Choube SC, Saket RK. Composite system reliability evaluation based on static voltage stability limit. J Inst Eng India, EL 2000;80(February):133–9. [7] Billinton R, Khan E. A security based approach to composite power system reliability evaluation. IEEE Trans Power Syst 1992;7(February): 65–70. [8] Pereira MVF, Pinto LMVG. A new computational tool for composite reliability evaluation. IEEE Trans Power Syst 1992;7(1):258–63. [9] Deng Z, Singh C. A new approach to reliability evaluation of interconnected power system including planned outages and frequency calculations. IEEE Trans Power Syst 1992;7(2):734–43. [10] Billinton R, Li W. A system state transition sampling method for composite system reliability evaluation. IEEE Trans Power Syst 1993;8(3):761–6. [11] Melo ACG, Mello JC, Granville S. Effects of voltage collapse problems in the reliability evaluation of composite systems. IEEE Trans Power Syst 1997;12(1):480–7. [12] Billinton R, Aboreshaid S. Voltage stability considerations in composite power system reliability evaluation. IEEE Trans Power Syst 1998;13(2):655–60. [13] Arya LD, Verma HK. A method for tracing PV-curve for voltage stability analysis with voltage dependent loads. Int J Electron Machines Power Syst 1996;24(July):583–96. [14] Billinton R, Allan RN. Reliability evaluation of engineering systems: Concepts and techniques. Pitman Books Limited; 1983.
L.D. Arya et al. / Electrical Power and Energy Systems 29 (2007) 99–105 [15] Billinton R, Allan R. Reliability evaluation of power systems. Plenum Press; 1984. [16] Bijwe PR, Kothari DP, Arya LD. Alleviation of line overloads and voltage violations by corrective rescheduling. Proc IEE C 1993;140(4):249–55. [17] Capitanescu F, Cutsem Thierry Van. Preventive Control of voltage security margins: A multi-contingency sensitivity based approach. IEEE Trans Power Syst 2002;17(2):358–64.
105
[18] Min Fu Li. Neural network in computer intelligence. Mc Graw Hill Inc; 1994. [19] Haykin S. Neural network: A comprehensive foundation. Pearson Education; 2003. [20] Zarate LALl, Castro CA, Ramos JLM, Ramos ER. Fast computation of voltage stability security margins using nonlinear programming techniques. IEEE Trans Power Syst 2006;21(1):19–27.
Probabilistic assessment and preventive control of voltage security margins using artificial neural network L.D. Arya a
a,*
, L.S. Titare b, D.P. Kothari
c
Electrical Engineering Department, Shri. G. S. Institute of Technology and Science, Indore (M.P.) 452003, India b Department of Electrical Engineering, Government Engineering College, Jabalpur, (M.P.) 482011, India c Centre for Energy Studies, I.I.T., Delhi, New Delhi 110016, India Received 18 February 2004; received in revised form 29 March 2006; accepted 15 May 2006
Abstract This paper describes a new technique for probabilistic assessment and preventive control of voltage security margins using artificial neural network. The probabilistic insecurity index (PISI) has been obtained for various operating conditions considering single and double line outages and accounting static voltage stability limit. Evaluated PISI has been obtained for various settings of reactive power control variables and loading conditions using cut-set method. These results have been used to train a multi layer feed forward network so as to assess the security on line. Further sensitivities of PISI have been evaluated based on trained network and have been used to control power system security. The algorithm has been implemented on a 6-bus, 7-line IEEE standard test system. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Voltage stability; Security; Probabilistic insecurity index; ANN
1. Introduction Power system security studies i.e. assessment and control forms an integral part of modern energy management systems (EMS). Power companies are facing a major challenge in maintaining quality and security of power supply due to ever increasing interconnections and loadings in the large power system networks. Economic constraint has forced the utilities to operate their generators and transmission systems to their maximum loadability point. Transient stability (synchronous) limits of the system have increased substantially with the development of improved protective devices, SVS, fast turbine valving and automatic voltage regulators. These developments have allowed to flow more real power over longer distances and systems which, used to be transient stability limited, are now facing the problem of maintaining the required bus voltages and *
Corresponding author. E-mail addresses: ldarya@rediffmail.com (L.D. Arya), lstitare@ yahoo.co.in (L.S. Titare), [email protected] (D.P. Kothari). 0142-0615/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2006.05.005
has become voltage stability limited [1]. Precisely voltage security has been defined as the ability of a system, not only to operate stably, but also remain stable (as far as the maintenance of the system voltage is concerned) following any reasonably credible contingency or adverse system change [2,3]. Further it has been a standard practice to evaluate reliability indices (probabilistic approach) based on security analysis accounting correctability, in planning stage [4–6]. If failure probability or as termed at times probability of load curtailment (PLC) is less, then robustness of the system is more towards violation of operating constraints. Evaluation of PLC requires large number of power flow studies under various loadings as well as outage conditions. Such type of index can not be used to assess security of the system in operating condition. Development of artificial neural network can make such failure probability indices applicable in real time applications in power systems. Most of the research work in evaluating the reliability indices did not account voltage security from voltage stability viewpoint. Billinton and Khan [7] used LP model for adequacy assessment of the composite power
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system accounts voltage and line flow constraints. MonteCarlo sampling with a variance reduction technique was developed by Pereira and Pinto [8] for reliability evaluation of a transmission system. A decomposition-simulation approach was used for power system reliability evaluation by Deng and Singh [9]. Billinton and Li [10] developed a system state transition sampling method for transmission system reliability evaluation. Melo et al. [11] and Billinton and Aboreshaid [12] accounted voltage stability constraints in evaluating the reliability indices. Melo et al. were probably the first to present an approach to calculate voltage collapse related indices as well as their impact on power system adequacy indices based on restoring system solvability by load shedding. Billinton and Aboreshaid developed a new methodology, which accounts voltage stability considerations in adequacy assessment of transmission network. In Ref. [6] a methodology has been developed for evaluating failure probability of a bulk power system based on peak load consideration and accounting correctability of the system. This has been achieved by optimizing static voltage stability limit using univariate search method and voltage stability limit have been obtained using a predictor–corrector based continuation power flow. Capitanescu and Cutsem [17] presented a multi contingency sensitivity based approach for preventive control of voltage security margins utilizing deterministic approach. In this paper a new viewpoint has been presented for voltage security assessment and control using reactive power rescheduling and if required even using load shedding. A probabilistic insecurity index (PISI) has been evaluated using cut-set method by considering single and double line outages. Voltage stability limits have been calculated using continuation power flows in each single and double line outage condition. The PISI have been obtained for various single and double line outages and for various reactive power control variables and loading conditions, using these results a multi layer ANN is trained to obtain PISI for any operating conditions. This trained network is then used to obtain sensitivities of PISI with respect to reactive power control variables and load sheds at various buses, as the input to network is reactive power control variables and load at each buses. For this closed form relations have been derived for sensitivities relating load-shed and reactive power control variables to PISI. These sensitivities are used for voltage securities control either using reactive power control variables and if necessary load shedding at selected buses. Application of ANN makes it suitable for on line applications. Next section describes the methodology for obtaining probabilistic insecurity index.
are directly related to the modes of system failure and therefore identify the distinct and discrete ways in which a system may fail. In identifying cut-set one identifies the most important line or combination of lines, which are of greatest importance. A minimal cut-set is a set of transmission line which, when failed, causes failure of the system but when any one element of the set has not failed, does not cause system failure. All component of a minimal cut-set must be in the failure state to cause system failure. Failure probability of a power network can be written as follows: P F ¼ P ½C 1 þ C 2 þ þ C n ;
where C1,C2,. . .,Cn are minimal cut-sets. This precise evaluation is always theoretically possible, but it is exhaustive and time-consuming exercise, which becomes prohibitive with power network. To overcome this problem, approximations are usually made in the evaluation of failure probability, which although reduce precision, permit much faster evaluation. The degree of inaccuracy introduced is usually negligible and within the tolerance associated with the data of the transmission line availability for power system, which have large values of repair rates and low value of failure rates for each transmission line. Hence, the failure probability is approximated by retaining first order terms in the expansion of Eq. (1) as follows: n X PF ¼ P ðC i Þ: ð2Þ i¼1
The second approximation is to neglect cut-sets of an order greater than a certain value. Number of components comprising that cut-set is equal to the order of a cut-set. In present paper cut-sets up to second order have been considered. This is due to the fact that probability of outage of more than two transmission lines is negligible. Further higher order cut-set under usual loading conditions may not be minimal cut-sets, which are required in Eq. (1). Hence relation (2) is utilized for evaluating probability of failure (PF) under above mentioned approximation. Errors in calculation of unreliability may even less than 1%. A two-state model (up state and down state) is used to model the operation of each transmission line. The up state indicates that the transmission line is in operating state and down state implies that the element is inoperable due to a failure or scheduled off. Fig. 1 shows the two-state model. In this figure ki and li are failure and repair rates of transmission line, respectively. This model is used to provide availability and unavailability of transmission line in long run expression for these are given as follows [14]:
2. Cut-set method and evaluation of probabilistic insecurity index The cut-set method is a powerful one for determining the reliability of a power network. The method can be easily programmed on a digital computer for fast and efficient solution of any general network. Moreover cut-sets
ð1Þ
λi Down state
Up state
μi Fig. 1. Two state model for each transmission line.
L.D. Arya et al. / Electrical Power and Energy Systems 29 (2007) 99–105
Ai ¼ li =ðki þ li Þ
ð3Þ
Ai ¼ ki =ðki þ li Þ
ð4Þ
Ai and Ai denote availability and unavailability of transmission line. Minimal cut-set (first and second order) for a specified load is obtained by comparing the load with static voltage stability limit in base case as well as under various line outage conditions. Static voltage stability limits under single or double line outage condition is obtained using continuation power flow methodology [13]. For adequate system operation sufficient stability margin must be maintained. Using the result of continuation power flow, one can prepare a capacity outage table, which gives permissible loading and corresponding state of the network. Now for various load levels one identifies cut-set from the capacity availability table. Relation (2) is utilized to get probability of failure. This probability of failure is termed as probabilistic insecurity index (PISI) of transmission system. It is further stressed that loadability limit depends on the settings of Perform base case load flow solution and obtain static voltage stability limit
Perform continuation power flow for single and double line outage condition (k) and obtain Plimit for each contingent condition
1
2
101
reactive power control variables. Hence failure probability or PISI is obtained for various settings of reactive power control variables. Computational sequences are shown in Fig. 2 with the help of a flow chart. In block 3–5 static voltage stability limit is obtained for various specified loads and cut-sets are obtained. In block-6 failure probability PF is calculated. In block-7 settings of reactive power control variable is altered in a mono variable fashion to generate PISI for various settings of control variables. 3. Evaluation of sensitivity of probabilistic insecurity index (PISI) with respect to reactive power control variables using artificial neural network Instances obtained in previous section are used to train a multi layer feed forward network. This network contains one input layer, one hidden layer and one output layer. The network is trained using back propagation algorithm [18]. Number of units in input layer equals to number of reactive power control variables and total number of load buses. Number of unit in output layer is one, which gives output as PISI. Further the neurons in the hidden layer are assumed to be sigmoidal. Neuron in output layer is assumed to be non-sigmoidal (linear). Network equations of Fig. 3 are written as follows: m X O¼ W jo Oj ðnon-sigmoidalÞ: ð5Þ j¼1
Output of jth neuron in hidden layer is as follows: Prepare capacity-outage probability table and contingencies indicating P(k) limit
Identify for various peak load consideration cut-sets C1(i), C2(i),...., Cr(i) for ith loading condition i = 1,...., NI
Calculate PF failure probability using eq.(9)
Change the control setting one by one in steps above and below up to limits Up = Up + ΔUp
3
Oj ¼ 1=ð1 þ eNetj Þ; where Netj is n X Netj ¼ W ij X i ;
4
j ¼ 1; 2; . . . ; m;
ð6Þ
i ¼ 1; . . . ; n:
ð7Þ
i¼1
In above equations Wjo are the weights connected between jth hidden neuron and output neuron. Wij are the weight Wij
5
U1
Σ
O1
U2
Σ
O2
W10 W20
6
U3
Σ
W30 O3
Σ
UNC
No
If all change have been made?
O (Output)
Wmo
P1
7 PNB
Σ
Om
Yes Input layer
Stop Fig. 2. Flow chart for computing probabilistic insecurity index.
Hidden layer (‘m’ units)
Output layer
Fig. 3. Generalised diagram of back propagation network for voltage security control and PISI calculations.
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connected between ith input node and jth hidden neuron. Xi is input variable at ith node [19]. Weight change DWjo are given by following formulae:
uled. Correction for such control variable is obtained as follows:
DW jo ¼ g d Oj ; d ¼ ðT OÞ;
where PISIO is the present insecurity index as obtained by the ANN, PISIth is the threshold value of the indicator and SPISr is the sensitivity of indicator with respect to rth control variable. If control correction as obtained using relation (14) violates the limit it can be put to limiting value. Then again modified sensitivities are evaluated and another control variable is selected for control action. This process of monovariable rescheduling terminates when PISI is brought within limits. At last load flow program is executed and it is verified that all voltages are within limits. If it is found that all reactive power control variables have been exhausted and even at this moment PISI violates the limit then sensitivities of PISI are evaluated with respect to load shed at the load buses and a bus is selected for load shed having highest sensitivity as follows:
ð8Þ ð9Þ
T and O are target value and output of network, respectively. ‘g’ is learning rate lies between [0, 1]. Expression for weight change DWij is given as follows using Back propagation algorithm: DW ij ¼ g dj X i ;
ð10Þ
where dj is error gradient and is given as follows: dj ¼ d:W jo :Oj ð1 Oj Þ;
ð11Þ
Xi is the element of input vector [X], where T
T
½X ¼ ½U 1 ; U 2 ; . . . ; U NC ; P 1 ; P 2 ; . . . ; P NB : Once the ANN of Fig. 3 is trained the network Eqs. (5)–(7) are used for calculating sensitivities of ‘O’ (PISI) with respect to input variables either control variables or disturbance variables (loads). Sensitivity expression is derived using chain rule of differentiation as follows: m X oPISI=oX i ¼ ½ðoPISI=oOj ÞðoOj =oNetj ÞðoNetj =oX i Þ j¼1
¼ SPISi ;
ð12Þ
where SPISi is the sensitivity of PISI with respect to Xi. Algebraic manipulation of network equations gives following expression for sensitivities: m X ½W jo Oj ð1 Oj ÞW ij : ð13Þ SPISi ¼ j¼1
Evaluation of sensitivities as governed by Eq. (13) is very simple and follows quickly from Fig. 3. Expression (13) can be introduced as sum of all path gains from an input node to output node. Path gain is the multiplication of all weights and gain of the nodes {Oj(1 Oj)} encountered in the path. This rule of calculating the sensitivity will be even applicable even for many hidden layers. The voltage security control algorithm is explained in next section. This algorithm also includes the possibility of load shedding.
DU r ¼ ðPISIO PISIth Þ=SPISr ;
ð14Þ
DP i ¼ ðPISIO PISIth Þ=SPIDi ;
ð15Þ
where SPIDi is sensitivity of PISI with respect to load shed at ith bus. This should not exceed a maximum amount of load shed at a bus (say 80% of total load at a bus, as 20% load is firm load and may be required for emergency services). Hence, if total load shed at a bus is insufficient, next bus is selected for load shed having highest sensitivity. This procedure continues till PISI is brought within limit. In a sense the ANN act as a system wise relay, which not only detects the insecure condition but also suggests for corrective action. This relay or smart software provides security to the system from voltage stability viewpoint. 5. Results and discussion The algorithm developed for voltage security control using rescheduling of reactive power control variables or load shed has been implemented on a 6-bus, 7-line IEEE standard test system [16]. Fig. 4 represents the test system. The system has two generating buses whose voltages can be adjusted between 0.95 6 Vi 6 1.15 pu for controlling the voltages of load buses. Availabilities of lines from line no.1 to line no. 7 are assumed as 0.99636033, 0.99636033,
1
4
3
2
4. A monovariable approach for voltage security enhancement Trained ANN is used for on-line assessment of voltage security and further sensitivities can be evaluated for corrective rescheduling. A threshold value of PISI can be selected for this purpose. At any instant, if the index PISI crosses the limit, evaluated sensitivities of this index with respect to control variables are used for corrective rescheduling. Control variable with highest sensitivity is resched-
6
5
Fig. 4. Line diagram of 6-bus system.
L.D. Arya et al. / Electrical Power and Energy Systems 29 (2007) 99–105
103
Table 1 Training instances for training ANN S. no
V1 (pu)
V2 (pu)
Pd3 (pu)
Pd4 (pu)
Pd5 (pu)
Pd6 (pu)
PISI as obtained by cut-set method
PISI as obtained by ANN
% Error
1 2 3 4 5 6 7 8
1.00 1.00 1.00 1.05 1.05 1.05 1.10 1.10
1.05 1.05 1.05 1.10 1.10 1.10 1.15 1.15
0.1925 0.4400 0.6050 0.4400 0.6325 0.7425 0.6050 0.8250
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.1050 0.2400 0.3300 0.2400 0.3450 0.4050 0.3300 0.4500
0.1750 0.4000 0.5500 0.4000 0.5750 0.6750 0.5500 0.7500
0.003446762 0.015166344 0.026306507 0.011581222 0.019627254 0.026306507 0.015166344 0.026306507
0.0031868 0.0159991 0.0246776 0.0110873 0.0211902 0.0270174 0.0148240 0.0264666
07.54 05.49 06.19 04.26 07.96 02.70 02.25 00.60
PISI as obtained using cut-set method and ANN.
0.99658703, 0.99636033, 0.99636033, 0.99658703, and 0.99658703. These typical figures have been taken from Ref. [15]. Probability of failure or PISI has been calculated for various settings of PV buses and loading conditions using the methodology of Section 2. Static voltage stability limits have been used for calculating PISI, which were obtained using predictor–corrector technique of Ref. [13]. Table 1 shows the values of PISI as calculated using cutset method and as obtained using trained network. This table also shows the percentage error in its last column. Table 2 gives various weights Wjo and Wij connected between hidden layer and output neuron and those between input layer and hidden layer. Numbers of neurons in hidden layer are six. Variables, put to input layer, are V1, V2, P3, P4, P5, and P6 i.e. voltages at generating buses and loads. Trained ANN has been validated with few testing instances other than used in training. The PISI as obtained using the trained network and cut-set method have been shown in Table 3 along with percentage errors for test cases. There is no rule for proportion between training and testing instances. In the present study training instances have been selected so as to encompass wide range of operating conditions for the power system. Further testing instances have been selected in this range, other than training instances. The trained network has been tested for 50 test instances. Only few (10) are given in Table 3 having highest magnitudes of errors. Values of probability of failure as termed PISI should not exceed a specified value (threshold value) say 0.01851. This value has been selected such that in all outage condition distance of voltage collapse point (in terms of MVA) is at least 20% of static voltage stability limit. This is an adequacy requirement. Depending on the operating philosophy and requirement of individual power utility this may vary. This threshold value is required in Eqs. (14) and (15) for evaluating control actions DUr or DPi. The probability of failure as obtained with trained ANN has percentage error less than 10%. If one want to reduce error than size of the ANN can be increased. This may be achieved by increasing number of neurons in hidden layer and increasing the hidden layers. Corrective rescheduling has been demonstrated for an operating condition
Table 2 Weights as obtained after training the back propagation network of Fig. 3 S. no
Weight between ith input node and jth hidden neuron (Wij)
W11 W12 W13 W14 W15 W16 W21 W22 W23 W24 W25 W26 W31 W32 W33 W34 W35 W36 W41 W42 W43 W44 W45 W46 W51 W52 W53 W54 W55 W56 W61 W62 W63 W64 W65 W66 W11 W21 W31 W41 W51 W61
0.15702530 0.14933560 0.03406646 0.03600000 0.20349430 0.09815425 0.24467800 0.27443130 0.06061339 0.10500000 0.05960826 0.04098819 0.10991660 0.02265394 0.10150800 0.17300000 0.03701656 0.01903466 0.27204240 0.22949380 0.20038590 0.18000000 0.08702040 0.22870280 0.04429901 0.06504500 0.02363667 0.07200000 0.00983370 0.09505729 0.08332255 0.04221593 0.09263467 0.18600000 0.13874920 0.21287110
Weight between jth hidden neuron and Oth output node (Wjo)
0.07430349 0.28554220 0.00087990 0.49452950 0.00048859 0.05809121
V1 = 0.95 pu, V2 = 1.00 pu and total load Pd = 1.7 pu. At this moment stability margin obtained is less than 20%. PISI
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Table 3 Validation of trained network using test cases as generated using cut-set method S. no.
V1 (pu)
V2 (pu)
Pd3 (pu)
Pd4 (pu)
Pd5 (pu)
Pd6 (pu)
PISI as obtained by cut-set method
PISI as obtained by ANN
% Error
1 2 3 4 5 6 7 8 9 10
0.95 0.95 1.00 1.00 1.05 1.05 1.05 1.10 1.00 1.10
1.00 1.00 1.05 1.05 1.10 1.10 1.10 1.15 1.05 1.15
0.1925 0.3850 0.3575 0.5775 0.5225 0.6875 0.4400 0.5500 0.5500 0.7150
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.1050 0.2100 0.1950 0.3150 0.2850 0.3750 0.2400 0.3000 0.3000 0.3900
0.1750 0.3500 0.3250 0.5250 0.4750 0.6250 0.4000 0.5000 0.5000 0.6500
0.00797143 0.018512031 0.011581222 0.022972705 0.015166344 0.022972705 0.011581222 0.011581222 0.022972705 0.019627254
0.0080719 0.0180551 0.0116980 0.0232249 0.0154400 0.0240996 0.0110874 0.0119379 0.0217746 0.0206276
01.25 02.46 01.00 01.09 01.54 04.90 04.26 03.08 05.21 05.09
Table 4 Voltages, loads before and after load shed for 6-bus test system S. no.
1 2 3 4 5 6
Bus voltages in pu
Load at buses in pu
Before load shed
After load shed
Before load shed
After load shed
1.1500 1.1500 0.8921 0.9127 0.8649 0.8834
1.1500 1.1500 0.9272 0.9526 0.9128 0.9446
0.0000 0.0000 0.9325 0.0000 0.5772 0.8251
0.0000 0.0000 0.9325 0.0000 0.5772 0.5171
on-line by training ANN using BPA,(iii) closed form relations have been derived to evaluate the sensitivity of PISI with respect to control variables and load shed and (iv) using the developed sensitivities, an algorithm has been developed for voltage security control and load shedding. The algorithm can be applicable on-line using the trained ANN. Since voltage instability is a phenomenon with slow dynamics [20]. Hence assessment and preventive actions are possible on-line using this ANN. References
obtained at this condition is 0.02238 pu. This is more than the specified threshold value. Sensitivities were evaluated at this moment. Voltage correction at bus No.1 is obtained using the sensitivity PISI with respect to this bus as DV1 = 0.076 pu. This brings PISI within limit (0.01851 pu) as final value. Another case taken at V1 = 1.15 pu, V2 = 1.15 pu at total load 3.30 pu, at this operating point PISI is 0.0283271 pu. It is observed at this point that correctability with reactive power reschedule is not possible. Hence sensitivities were evaluated with respect to load shed at the various load buses. The bus no. 6 having highest sensitivity 0.0319595 have been selected for load shedding. The amount of load shed as calculated using relation (15) is 0.307905 pu is within limit. The PISI calculated as 0.01850 pu, which is less than the threshold value. Voltages, loads before and after load shed are given in Table 4. 6. Conclusions An algorithm with a new viewpoint has been presented for voltage security assessment and control based on probability of failure of a system. The probabilities of failure for various operating conditions have been obtained using cutset method accounting static voltage stability limit. These cases have been used to train ANN using BPA. Further BPA has been used to obtain sensitivity of PISI with respect to control variables and load shed to maintain at least the threshold value of PISI. Main contributions of this paper are (i) development of an algorithm for evaluating PISI using cut-set method accounting static voltage stability limit,(ii) the applicability of PISI has been made
[1] Taylor CW. Power system voltage stability. Mc Graw Hill Inc; 1994. [2] Concordia C. Voltage instability. J Elec Power and Energy Syst 1991;13(1):14–24. [3] IEEE System Dynamic Performance Subcommittee. Voltage stability of power systems concepts, Analytical tools and industry experience. IEEE document 90th 0358-2-PWR-1990. [4] Sebastian Rios M, Victor Vidal P, Kiguel David L. Bus based reliability indices and associated costs in the bulk power system. IEEE Trans Power Syst 1998;13(3):719–23. [5] Billinton R, Wenyuan L. Hybrid Approach for reliability evaluation of composite generation and transmission systems using Monte-Carlo simulation and enumeration technique. IEE Proc C 1991;138(3): 233–41. [6] Arya LD, Choube SC, Saket RK. Composite system reliability evaluation based on static voltage stability limit. J Inst Eng India, EL 2000;80(February):133–9. [7] Billinton R, Khan E. A security based approach to composite power system reliability evaluation. IEEE Trans Power Syst 1992;7(February): 65–70. [8] Pereira MVF, Pinto LMVG. A new computational tool for composite reliability evaluation. IEEE Trans Power Syst 1992;7(1):258–63. [9] Deng Z, Singh C. A new approach to reliability evaluation of interconnected power system including planned outages and frequency calculations. IEEE Trans Power Syst 1992;7(2):734–43. [10] Billinton R, Li W. A system state transition sampling method for composite system reliability evaluation. IEEE Trans Power Syst 1993;8(3):761–6. [11] Melo ACG, Mello JC, Granville S. Effects of voltage collapse problems in the reliability evaluation of composite systems. IEEE Trans Power Syst 1997;12(1):480–7. [12] Billinton R, Aboreshaid S. Voltage stability considerations in composite power system reliability evaluation. IEEE Trans Power Syst 1998;13(2):655–60. [13] Arya LD, Verma HK. A method for tracing PV-curve for voltage stability analysis with voltage dependent loads. Int J Electron Machines Power Syst 1996;24(July):583–96. [14] Billinton R, Allan RN. Reliability evaluation of engineering systems: Concepts and techniques. Pitman Books Limited; 1983.
L.D. Arya et al. / Electrical Power and Energy Systems 29 (2007) 99–105 [15] Billinton R, Allan R. Reliability evaluation of power systems. Plenum Press; 1984. [16] Bijwe PR, Kothari DP, Arya LD. Alleviation of line overloads and voltage violations by corrective rescheduling. Proc IEE C 1993;140(4):249–55. [17] Capitanescu F, Cutsem Thierry Van. Preventive Control of voltage security margins: A multi-contingency sensitivity based approach. IEEE Trans Power Syst 2002;17(2):358–64.
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[18] Min Fu Li. Neural network in computer intelligence. Mc Graw Hill Inc; 1994. [19] Haykin S. Neural network: A comprehensive foundation. Pearson Education; 2003. [20] Zarate LALl, Castro CA, Ramos JLM, Ramos ER. Fast computation of voltage stability security margins using nonlinear programming techniques. IEEE Trans Power Syst 2006;21(1):19–27.