International Journal of Electrical Engineering Education  Transmission line shunt and series compensation with voltage sensitive loads Ulas Eminoglu  M
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International Journal of Electrical Engineering Education Transmission line shunt and series compensation with voltage sensitive loads Ulas Eminoglu M

Hakan Hocaoglu and Tankut Yalcinoz Department of Electronics Engineering Gebze Institute of Technology Kocaeli Turkey Department of Electrical and Electronics Engineering Nigde University Nigde Turkey Department of Electrical and Electronics Enginee

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International Journal of Electrical Engineering Education Transmission line shunt and series compensation with voltage sensitive loads Ulas Eminoglu M




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International Journal of Electrical Engineering Education 46/4 Transmission line shunt and series compensation with voltage sensitive loads Ulas Eminoglu , M. Hakan Hocaoglu and Tankut Yalcinoz Department of Electronics Engineering, Gebze Institute of Technology, Kocaeli, Turkey Department of Electrical and Electronics Engineering, Nigde University, Nigde, Turkey Department of Electrical and Electronics Engineering, Meliksah University, Talas, Kayseri, Turkey E-mail: hocaoglu@gyte.edu.tr Abstract This paper presents an analysis of the effects of shunt and series line

compensation levels on the transmission line voltage pro le, transferred power and transmission losses for different static load models. For this purpose, a simple model is developed to calculate the series and/or shunt compensated transmission line load voltage. Consequently, different shunt and series compensation levels are used with several voltage sensitive load models for two different line models. It is observed that the compensation level is signifi cantly affected by the voltage sensitivities of loads. Moreover, the voltage level of the transmission is an important issue for the

selection of the shunt and series capacitor sizes when load voltage dependency is used. Keywords selection of capacitor sizes; shunt and series compensations; transmission systems; voltage sensitive loads In electrical power systems, shunt capacitive compensation is widely employed to reduce the active and reactive power losses and to ensure satisfactory voltage levels during excessive reactive loading conditions. Shunt capacitive compensation devices are normally distributed throughout transmission or distribution systems so as to minimise losses and voltage drops. There are two types of

shunt compensa- tion: active and passive. For passive compensation, shunt capacitors have been extensively used since the 1930s. They are either permanently connected to the system, or switched, and they contribute to voltage control by modifying character- istics of the network. Improvements in the eld of power electronics have had a major impact on the development of shunt active compensators, which are Static Var Compensator (SVC) and Static Compensator (Statcom) devices. One of the most important applications of such devices is to keep system voltage pro les at desirable levels by

compensating for the system reactive power. By employing these devices for reactive power compensation, both the stress on the heavily loaded lines and losses are easily reduced as a consequence of line loadability, which is increased. In series compensation, capacitors are connected in series with the transmission and distribution lines. This reduces the transfer reactance between buses to which the line is connected, increases the maximum power that can be transmitted, and reduces the effective reactive power losses. Although series capacitors are not usually implemented for voltage control,

they do contribute to improving the system voltage and reactive power balance. The reactive power produced by a series capaci- tor increases with transferred power of the transmission line. 3–4
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Shunt and series line compensation 355 International Journal of Electrical Engineering Education 46/4 In electrical power systems, load modelling is a dif cult problem due to the fact that the electrical loads of a system comprise residential, commercial, industrial and municipal loads. It should also be noted that variation of the loads over time and number of uncertainties, spanning

from economic parameters to the weather condi- tions, signi cantly increase the complexity of the load modelling process. On the other hand, aggregate load models, which represent the load as an algebraic equation, have extensively been used for various power system studies to understand and analyse system behaviour under various conditions. Traditionally, most of the con- ventional load ow methods, for transmission and distribution systems, use the constant-power load model. The constant-power load model is highly questionable, especially for a distribution system where most of the buses are

uncontrolled. For transmission systems, where loads are generally served through transformers equipped with OLTCs (on-load tap changers), it is reasonable to use a constant power model for the analyses. However, economic and environmental force con- strains the system operators to exploit existing power structure to the limits. This can cause voltage stability problems and increases the risk of voltage collapse. It is widely recognized that, for weak power systems, the dynamic behavior of OLTCs contributes to the voltage collapse. Accordingly, OLTC blocking becomes a usual practice among the

operators of weak systems. Therefore, like distribution systems, the constant power load model becomes questionable for particularly weak transmis- sion systems. Accordingly a number of studies, found in the literature, deal with the effects of static load models on various power systems phenomena. 7–15 In Ref. 7, the authors analysed the effect of static loads modelled as an exponential form on the optimal load ow solution. The load ow solutions are compared with the standard optimal load ow solutions. The authors showed that the differences in fuel cost, total power loss and voltage values

are signi cant. Moreover, they con- cluded that the required iteration numbers are higher when the system is heavily loaded. For distribution systems with constant-power, constant-current and constant impedance loads, a new load ow algorithm has been proposed and the effects of these load models on the convergence pattern of the load ow method have been studied by Haque in Ref. 8. Results of the load ow show that the constant power load model gives the lowest voltage pro le while the constant impedance load model provides the highest voltage pro le. It is seen from the results that the

convergence of the load ow solutions gets dif cult when load exponents increase. The effects of voltage-dependent load on the convergence ability of the load ow method for different characteristics of the distribution system are also analysed in Ref. 9. In this study, the convergence ability of the proposed power ow algorithm has been com- pared with the Ratio-Flow method, 10 which is based on Kirchoff Voltage Law (KVL) and Kirchoff Current Law (KCL) for different loading conditions, different R/X ratio and different voltage levels, under a wide range of load exponents in radial distribu- tion

systems. The authors have concluded that load exponents have a signi cant effect on the convergence ability of KVL- and KCL-based load ow method and load ow solutions. The effect of shunt capacitor compensation on the voltage regulation of distribu- tion systems for different static load models has been presented. 12 A set of non-linear
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356 U. Eminoglu, M. H. Hocaoglu and T. Yalcinoz International Journal of Electrical Engineering Education 46/4 equations is established for radial systems by considering power balance and injected power in terms of system parameters;

consequently these equations are solved for three types of loads: constant power, constant current and constant impedance load. In this study, the effects of shunt capacitor levels on load voltage magnitude are analysed for voltage levels lower than 1 p.u. It is demonstrated that the effect of shunt capacitor sizes on the voltage magnitude increases with decrease of voltage sensitivity of the static load. El-Metwally 13 has developed a method for assessing the loadability limit of high voltage compensated transmission lines taking into account the effect of load characteristics. The effect of

voltage and/or frequency- dependent load on the maximum power transfer limit and critical voltage of the series and shunt compensated lines are investigated. Results show that the critical voltage and maximum power transfer limit of the less voltage-sensitive loads are greater than the more sensitive load. Ramalingam and Indulkar 14 have presented the effects of load characteristics on the load voltage and current magnitudes, load phase angle, active and reactive power losses of transmission lines for different static load types. Results show that when load voltage sensitivity increases, the

transferred power and the transmission line power loss decrease. The same authors 15 have also analysed the effects of tap-changing transformer control on the power voltage char- acteristics of compensated EHV transmission lines for different static load types. It should be noted that authors have only studied under 1 p.u. line voltage level. The effect of high voltage level is not studied in all references cited above. This paper presents the effects of shunt and series compensation levels on the transmission system voltage pro le, transferred power, and line losses for different static load

models. For this purpose a simple model is developed to calculate series and/or shunt compensated transmission line load voltage. The developed theory takes into account voltage dependency of static loads, transmission line parameters, and series and/or shunt compensator reactance. Two different line models (nominal circuit model and distributed line model) are used for analysing the effects of dif- ferent static load models on transmission system performance. Effect of voltage sensitivity of loads on the selection of shunt capacitor sizes for different voltage levels is also analysed giving

particular emphasis to the voltage level higher than 1 p.u. Static load models In electric power system analysis, loads may be modelled as a function of voltage and/or frequency and this type of modelling is considered static. Common static load models for active and reactive powers are expressed in a polynomial or an exponen- tial form. A static load model depending on the powers relation to the voltage as an exponential equation is given by PP np (1) QQ nq (2)
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Shunt and series line compensation 357 International Journal of Electrical Engineering Education 46/4 Where and are

the active and reactive load powers, respectively, at the nominal voltage of and is the actual voltage magnitude in p.u. The parameters np and nq stand for the voltage sensitivities of the static load. The polynomial load model is a static load model that represents the power- voltage relationship as a polynomial equation of voltage magnitude. It is usually referred to as the ZIP model and consists of a combination of three different expo- nential load models, namely constant impedance ( ), constant current ( ) and con- stant power ( ). The active and reactive power characteristics of the ZIP

load models are given by: PPa a oo oo =+ 12 (3) QQb b oo oo =+ 12 (4) Where and are the parameters for constant power load component; and are the parameters for constant current load component; and are the parameters for constant impedance load component. The values of these coef cients are determined for different load types in transmis- sion and distribution systems. Usually data, determined from experience, could be used for the estimation. Common values for the exponents of static load model for different load components are widely analysed in the literature. These static loads may have

high voltage dependency such as battery charge or televisions. 16 The aggregate load model may attain high values of exponents in the system nodes where the proportion of such devices is signi cant. These exponents may be valid for only a limited voltage range, which are 10% of the 1 p.u. voltage level. At very high and low voltage magnitude levels the models are inadequate for some load types such as motors and lighting. 16–17 Transmission line compensation Transmitted real and reactive power of the transmission line, which is shown in Figs 1(a) and (b), can be derived in terms of the ABCD

parameters of line using the following notation: VV AV rS Bsr BA =−+ cos cos (5) VV AV rS Bsr BA =−+ sin sin (6) Where and stand for phase voltages at bus s and bus r, and are transmis- sion line parameters of phase a. and stand for phase angles of the parameter
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358 U. Eminoglu, M. H. Hocaoglu and T. Yalcinoz International Journal of Electrical Engineering Education 46/4 Fig. 1 The circuit models of a transmission line. (a) Nominal TT-circuit model; (b) distributed parameters model. and , and and for the voltage of bus s and bus r. The value of and changes

according to the line model. Equations (5) and (6) can be rewritten as: cos cos Bsr sr BA PB VV AV −+ =+ (7) sin sin Bsr sr BA QB VV AV −+ =+ (8) by using the trigonometric identity cos sin 22 Bsr Bsr −+ +−+ (9) and substituting eqns (1)–(2) and (7)–(8) into eqn (9), the polynomial equation of the load voltage for the exponential load model is obtained as follows: AV AV BPV QV VV rror np BA or nq BA sr 42 22 + + () cos sin VQVB np or nq 222 () (10) Equation (10) has a straightforward solution and depends on the voltage dependency terms of exponential static load model

and line parameters. It is noted that from the solutions for that only the highest positive real root of this equation is used in the analyses. When polynomial load model is used, the voltage equation can be obtained by using eqns (3) and (4) instead of eqns (1) and (2), and then the function of the load voltage can be written as: AV AV BPa aV aV Qb bV bV rroorrBA oo r r 42 12 12 +++ () +++ () cos sin BA sr oo r r oo r r VV P a aV aV QbbVbV ()) −+ ++ () +++ 22 2 12 12 (11) Shunt compensation Shunt compensation is used by utilities at both transmission and distribution levels. In the

shunt compensation applications, passive shunt capacitors are extensively
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Shunt and series line compensation 359 International Journal of Electrical Engineering Education 46/4 used for power factor correction and to improve the system voltage pro le by eco- nomical means. The principal advantages of shunt capacitors could be listed as their low cost, exibility on installation and practical operations. To show the effect of shunt compensation on the system voltage pro le, a simple two-bus system given in Fig. 1 is used and a shunt capacitor is added.. The effect of shunt

compensation can be introduced using Thevenin’s theorem by keeping the load phase angle at a constant value. For this case, the Thevenin equivalent voltage ( th ) and impedance th ) seen from the end of the line (point of r) can be written for each line model, given in Fig. 1, by neglecting the active losses of the shunt capacitor. When the nominal circuit model of a transmission line, given in Fig. 1(a), is used the th and th seen from the end of the line (point of r) can be written as: ZrjX th ++ () (12) Zr jX ZrjX th ++ (13) Zj XX XX CSh CSh = (14) Sh Sh (15) When the distributed model of a

transmission line, given in Fig. 1(b), is used, the cascaded system parameters and th seen from the end of the line (point of r) with the shunt compensation ( jX ) can be written as: AjBX BjAX CjDX DjCX CC CC + + (16a) when there is no current on the receiving end, the solution of the receiving end voltage of eqn (16a) gives the Thevenin equivalent voltage as: AjBX th (16b) and, similarly th seen from the end of the line (point of r) can be written by using the ratio of open circuit voltage and short circuit current as: jBX BjAX th −+ (17) where is the line series reactance; is the line

series resistance; is the line sending end voltage magnitude; Sh is the line shunt capacitor admittance;
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360 U. Eminoglu, M. H. Hocaoglu and T. Yalcinoz International Journal of Electrical Engineering Education 46/4 is the shunt capacitor reactance; and are parameters of the transmission line. The Thevenin equivalent voltage ( th ) and equivalent impedance ( th ) increase with the shunt compensation levels by causing an increase on the load voltage magnitude. The effect of shunt capacitor on the load voltage can be analysed by substituting th and th into eqns (10) and (11)

instead of and . When the Thevenin equivalent circuit is used, load voltage varies with shunt capacitor reactance, line parameters and load nominal power. In this case, the parameters of each transmission line are chosen as 1 in eqns (10) and (11). Series compensation Series capacitive compensation in a.c. transmission systems can yield several bene- ts such as increases in power transfer capability and enhancement in transient sta- bility. For the series compensation, series capacitors are connected in series with the line conductors to compensate the inductive reactance of the line. This

reduces the transfer reactance between buses to which the line is connected, increases maximum power that can be transmitted, and reduces effective reactive power loss. Although series capacitors are not usually installed for voltage control, they do contribute to improving the voltage pro le of the line. When the nominal circuit model of a transmission line, which is given in Fig. 1(a), is used, the effect of series compensation on the transmission line voltage pro le, transferred power and line losses can be analysed by using eqn (18) in the calculation of the parameters of and given in eqn

(10) for exponential load model and in eqn (11) for polynomial load model. zrjxx =+ (18) Where is the line series reactance, is the series capacitor reactance, and is the line series resistance. In addition, the effect of series compensation on the transmis- sion line voltage pro le can be analysed by using the ABCD parameters of the series capacitor reactance when a distributed parameters line model is used. In this case, the equivalent ABCD parameters of the compensated line can be calculated by using matrix multiplication of cascaded line and series capacitor parameters. Analysis of shunt

and series compensations A two-bus power system is used and analysed for different static load models with different shunt and series capacitor sizes. The two-bus system parameters are selected as 1.1 p.u., 7.0 p.u. and 1.0 p.u. with the base of 100 MVA and 345 kV. The load exponents are being varied from 0 to 5 in order to show the effect of load exponents on the transmission system performance. Two transmission line models are used. The line parameters are selected as Z 0.032 0.35 /km, sh 4.2 * 10 S/km and l 200 km. Analysis of shunt compensation Five different capacitor sizes are employed

for the shunt compensation with different voltage-sensitive load models and their voltage magnitudes obtained by solutions of
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Shunt and series line compensation 361 International Journal of Electrical Engineering Education 46/4 eqn (10). Variations of the load voltage magnitude with different compensation levels are given in Fig. 2 for two line models and in Fig. 3 for the nominal model of a transmission line. From Figs 2 and 3, it is seen that shunt compensation has approximately the same effect on the load magnitude for high values of exponents of static load models. On the

other hand, the effect of shunt compensation on the voltage pro le for low values of the exponents is more signi cant than the high values of the exponents. From these gures it is seen that load voltage magnitude varies with the load model, with the increase in shunt capacitor size being highest for the system when load voltage dependency decreases. It should be noted that for a load voltage magnitude at 1 p. u., the load exponents have no effect on the shunt compensation level, as can be seen from eqn (10). Moreover, the nominal circuit model has a lower voltage level than the distributed

parameters line model for each load model and the slopes change with the load model with different shunt capacitor size being the same for each line model. Required reactive powers to keep the load voltage at predetermined levels are presented in Fig. 4 for two line models and Fig. 5 for a nominal circuit line model. Fig. 2 Variation of the load voltage magnitude (p.u.).
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362 U. Eminoglu, M. H. Hocaoglu and T. Yalcinoz International Journal of Electrical Engineering Education 46/4 As depicted in gures, load models have a signi cant effect over the required com- pensation power.

If load exponents decrease, the required reactive power increases when the voltage level is lower than 1 p.u. On the other hand, if load exponents decrease, the required reactive powers decrease when the voltage level is higher than 1 p.u. From Fig. 4 it is seen that to keep the load voltage at 0.96 p.u. for the nominal and distributed parameters line models, 33.41 and 31.57 MVAr capacitors would be needed in the case of a constant power load, whereas 2.67 and 2.61 MVAr capaci- tors are required if the load is modelled as a constant impedance load respectively and for each line model the

magnitude of required reactive powers decrease when load exponents increase. On the other hand, to keep the load voltage at 1.04 p.u., 127.28 and 125.59 MVAr capacitors are required if the load is modelled as a constant power load, whereas 162.38 and 160.5 MVAr capacitors are required if the load is modelled as a constant impedance load, and required capacitor is on the increase with the load exponents. Moreover, for different shunt compensation levels, varia- tions of load active and reactive powers and transmission line losses, given in Fig. 6, with different voltage sensitivities of

different static load models are analysed when the nominal circuit model of the transmission line is used. It is seen from the gure that transmission line voltage level has more effects on the line losses when a different static load model is used. The results are summarised as follows: Fig. 3 Variation of the load voltage magnitude (p.u.).
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Shunt and series line compensation 363 International Journal of Electrical Engineering Education 46/4 Results show that transferred active power and reactive power increase with increase of compensation level for each load type due to

increase in load voltage magnitude. It is clearly seen that for the load which has low voltage sensitivities, the transferred active and reactive power attain higher values than the load with high voltage sensitivities, for each compensation level when the voltage magni- tude is lower than 1 p.u., which concurs with the previous ndings. On the other hand, transferred powers have higher values for more sensitive loads when voltage magnitude is higher than 1 p.u. From Fig. 6, active power losses are higher for the low voltage sensitive loads than high voltage sensitive loads, for each

compensation level when voltage magnitudes are lower than 1 p.u. On the other hand, power losses are higher in the case of high voltage sensitive loads than in the case of low voltage sensitive loads when voltage magnitudes are higher than 1 p.u. As expected, voltage sensitivity of loads has no effect on the transferred power and transmission losses for the load voltage magnitude of 1 p.u. Fig. 4 Variation of required reactive power for different load voltage levels (MVAr).
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364 U. Eminoglu, M. H. Hocaoglu and T. Yalcinoz International Journal of Electrical Engineering

Education 46/4 Analysis of series compensation The effect of series line compensations on voltage pro le, transferred power, and power losses of transmission systems for exponential static load models is analysed for the same system for two transmission line models by locating the series capacitor in the middle of the line. Transmission line series reactance is compensated at 10%–60% ratios to show the effects of load characteristics on the load voltage pro le with different series compensation levels. Results are given in Fig. 7 for two line models and in Fig. 8 for only the nominal line

model. As illustrated in Figs 7 and 8, the effect of series compensation on voltage pro le for low values of the exponents is more signi cant than high values of exponents. The slopes change with the load model, with the increase of series compensation levels being highest for the system with decrease of values of exponents. For a xed load voltage, it is clearly seen that different voltage sensitive loads require different series capacitor sizes and capacitor sizes increase with decreasing voltage sensitivi- ties of loads. Nominal circuit line model has higher voltage level than distributed

parameter line model for ratios of 0.0%–40% series line compensation. However these differences in voltage magnitudes decrease with the increase in the line series Fig. 5 Variation of required reactive power (MVAr).
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Shunt and series line compensation 365 International Journal of Electrical Engineering Education 46/4 Fig. 6 Variation of transmission line losses (p.u.). Fig. 7 Variation of the load voltage magnitude (p.u.).
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366 U. Eminoglu, M. H. Hocaoglu and T. Yalcinoz International Journal of Electrical Engineering Education 46/4 compensation levels. It is

seen that the nominal circuit model overestimates the voltage when line reactance is compensated less than 50% for given parameters set. On the other hand, the model underestimates the voltage magnitude when line reac- tance is compensated more than 50%. Overestimation attains a maximum 1.8% of 1 p.u. and gradually decreases when the compensation level increases. Moreover, the line model affects only the voltage magnitude of the load and the slopes change with the load model, with different series capacitor size being the same for each line model. For different series compensation levels,

variations of transferred active and reac- tive powers, and transmission losses with different voltage sensitivities of load models are analysed and transmission losses are given in Fig. 9 in the case of using the nominal line model. Results show that, when the load voltage is below the magnitude of 1 p.u., transferred active and reactive powers, and transmission losses increase with decreasing load exponents in the case of the ratio of 0% (no compen- sation)–20% series line compensations. On the other hand, the ratio of 30%–60% series line compensations causes voltage levels higher than 1

p.u. for this particular transmission system. Thus, transferred power and transmission losses increase with load exponents. From these results, it can be said that load characteristics have a Fig. 8 Variation of the load voltage magnitude (p.u.).
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Shunt and series line compensation 367 International Journal of Electrical Engineering Education 46/4 signi cant effect on the transmission line performance and a selection of series compensation levels. Conclusions This paper presents an analysis of the effects of shunt and series compensation levels on the transmission system

voltage pro le, transferred power, and line losses for different static load models. The voltage expressions, which are derived for different load models, depend on load nominal active and reactive power, voltage sensitivities of load and line parameters. Then, different shunt and series compensation levels were used for shunt and series compensation with several voltage sensitive load models by using two different transmission line models. Results show that voltage sensitivities of load have signi cant effects on receiving end voltage magnitude of the line, transferred power and power losses

for different sizes of series and shunt capacitors. It is evident that load exponents have a signi - cant in uence on the required shunt capacitor sizes. Effects of shunt and series capacitors on the load voltage increase when voltage sensitivities of loads decrease. It is concluded that for a xed load voltage level, the required shunt compensation Fig. 9 Variation of transmission line losses (p.u.).
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368 U. Eminoglu, M. H. Hocaoglu and T. Yalcinoz International Journal of Electrical Engineering Education 46/4 levels decrease with increasing voltage sensitivity of the loads

below a voltage magnitude of 1 p.u. On the other hand, required shunt compensation levels increase with voltage sensitivity of loads above the voltage magnitude of 1 p.u. for the 1 p.u. voltage magnitude, voltage sensitivities of loads are not important and all loads require the same amount of shunt capacitor size. The voltage dependency of loads is the primary factor in the determination of series and shunt capacitor sizes. Therefore the nature of the served load in terms of dependence on the voltage should be known and taken into account. The nominal circuit model has a lower voltage level

than the distributed parameters line model for each load model in the case of shunt compensation. It is seen that the nominal circuit model overestimates the voltage magnitude when line reactance is compensated at low levels (less than 1 p.u.). On the other hand the model underestimates the voltage magnitude when line reactance is compensated at high level. These differences of voltage magnitude decrease with increasing line series compensation levels and the line voltage reaches a higher value than the voltage magnitude of the nominal line model when line series reactance is compensated at a

high ratio. Slopes change with the load model, with different shunt and series capacitor sizes being nearly the same for each line model. References 1 P. Kundur, Power System Stability and Control (McGraw-Hill, New York, 1994), pp. 627–633. 2 Y. H. Song and A. T. Johns, Flexible AC Transmission Systems (FACTS) (IEE, Stevenage, 1999), pp. 146–166. 3 E. C. Starr and R. D. Evans, ‘Series capacitors for transmission circuits’, AIEE Trans. , 61 (1942), 963. 4 B. S. Ashok Kumar, K. Parthasarathy, F. S. Prabhakara and H. P. Kincha, ‘Effectiveness of series capacitors in long distance transmission

lines’, IEEE Trans ., 89 (1970), 941–950. 5 H. Ohtsuki, A. Yokoyama and Y. Sekine, ‘Reverse action of on-load tap changer in association with voltage collapse’, IEEE Trans. Power Systems , (1) (1991), 300–306. 6 T. X. Zhu, S. K. Tso and L. K. Lo, ‘An investigation into the OLTC effects on voltage collapse’, IEEE Trans. Power Syst ., 15 (2) (2000), 515–521. 7 L. G. Dias and M. E. El-Hawary, ‘Effects of active and reactive power modeling in optimal load ow studies’, IEE Proc. , 136 (5) (1989), 259–263. 8 M. H. Haque, ‘Load ow solution of distribution systems with voltage-dependent load models’,

Electrical Power Syst. Res. , 36 (1996), 151–156. 9 U. Eminoglu and M. H. Hocaoglu, ‘A new power ow method for radial distribution systems includ- ing voltage dependent load models’, Electrical Power Syst. Res ., 76 (2005), 106–114. 10 J. Liu, M. M. A. Salama and R. R. Mansour, ‘An ef cient power ow algorithm for distribution systems with polynomial load’, Int. J. Elect. Enging Educ. , 39 (4) (2002), 372–386. 11 IEEE Task Force on Load Representation for Dynamic Performance, ‘Bibliography on load models for power ow and dynamic simulation’, IEEE Trans. Power Syst ., 10 (1995), 525–538. 12 M.

Mithulananthan, M. A. Salama, C. A. Canizares and J. Reeve, ‘Distribution system voltage regulation and VAr compensation for different static load models’, Intl. J. Elect. Enging Educ. , 37 (4) (2000), 384–395. 13 M. M. El-Metwally, A. A. El-Emary and M. El-Azab, ‘Effect of load characteristics on maximum power transfer limit for HV compensated transmission lines’, Elec. Power Energy Syst ., 26 (2004), 467–472. 14 K. Ramalingam and C. S. Indulkar, ‘Transmission line performance with voltage sensitive loads’, Int. J. Elect. Enging Educ. , 41 (1) (2004), 64–70.
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line compensation 369 International Journal of Electrical Engineering Education 46/4 15 K. Ramalingam and C. S. Indulkar, ‘Effect of on-load tap-changing transformer control on power voltage characteristics of compensated EHV transmission lines with voltage-sensitive loads’, J. Inst. Engineers (India) , 84 (2004), 221–226. 16 T. V. Cutsem and C. Vournas, Voltage Stability of Electric Power Systems (Kluwer, Dordrecht, 1998), Ch. 4. 17 C. W. Taylor, Power System Voltage Stability (McGraw-Hill, New York, 1994), pp. 17–135.
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