Extended Transmission Line Loadability Curve by Including Voltage Stability Constrains

Transcription

1 Extended Transmission Line Loadability Curve by Including oltage tability Constrains Jin Hao, Member, IEEE, and Wilsun Xu, Fellow, IEEE Abstract t. Clair curve provides a simple means for estimating power transfer capabilities of transmission lines. It concerns three limiting factors: thermal limit, voltage quality (or drop) limit, and angular stability limit. This paper illustrates the influence of voltage stability limit and presents an extended loadability curve. Moreover, the impacts of line resistance and shunt compensation on line loadability are investigated. Index Terms Transmission line loadability, t. Clair curve, voltage stability limit, surge impedance loading. T I. INTODUCTION ransmission line loadability curve, also known as t. Clair curve [] has been a valuable tool for quickly estimating the power transfer capabilities of transmission lines. Due to its universal characteristics, i.e. applicable to all voltage levels, t. Clair curve are generally accepted in the industry as a convenient reference for estimating the maximum loading limits on transmission lines. The t. Clair curve [],[], presented in Fig., shows the loadability of transmission line in terms of their urge Impedance Loading (IL). It is well known that the per-unit line data normalized using IL and urge Impedance is constant, i.e. independent of line construction and voltage rating. Therefore, this curve can be used universally. Line load limit in pu of IL Fig.. Transmission line loadability curve (t. Clair curve) [] In recent years, power system voltage stability has attracted considerable interest in industry. Therefore, it is important to include the voltage stability limit in the line loadability curve. This paper illustrats the influence of voltage stability limit on line loadability. The impacts of line resistance and shunt compensation are also investigated. Besides, the voltage drop limit and voltage stability limit are compared for different transmission lines. II. EXTENDED TANMIION LINE LOADABILITY CUE In this section, we first introduce the basic concepts of voltage stability limit and then present the extended transmission line loadability curve with considering the voltage stability limit. A. oltage tability Limit A simple system is shown in Fig.. In the system, to simplify the calculation, the voltage phase angle at the receiving end is seen as reference, and the voltage magnitude of the sending end,, is constant. jx Fig.. ystem diagram At the receiving end, * I jq o I +jq () jq I () The sending end voltage is: jq QX X jx j The corresponding magnitude equation is () J. Hao and W. Xu are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, TG, Canada ( jhao@ece.ualberta.ca, wxu@ualberta.ca ).

2 QX X The power delivered to the load as a function of receiving end voltage when Q = can be solved as: () X ince is constant and close to per-unit, and X can not change. is the only variable that can vary. o the power will vary with, which is shown in Fig.. Operating point max () It can be seen from the equivalent circuit, when the line length increases, the open circuit voltage -eq increases accordingly because of the line charge. This is the well-known Ferranti effect. This effect leads to the increase of nose point voltage nose. We can further deduct from the figure that when the length increases to a specific value, the nose point voltage will become higher than the sending end voltage. When this happens, it becomes impossible to operate the system as any operating point with acceptable voltage level will be below the nose point, which is an unstable case. The line length at which the nose point will move above the sending end voltage can be determined using the following condition: nose (8) assuming that the receiving end Q load is equal to, we can establish eq nose (. L) / olving the above equation yields: (9) Fig.. Transfer capability curve L 88.7km () (.) The maximum power that can be transmitted is reached when d/d =, which can be determined as max_ stab () X The voltage corresponding to () is nose (7) The above limit () is called the oltage tability Limit of power transmission and the subscript stab is used for this consideration. If we use the nominal I circuit to approximate the line, the transmission scheme and its equivalent circuit are shown in Fig.. eq s (. L) / jx eq Z=j.L Y=j./*L A) Nominal I circuit of a transmission line j.l (. L) / B) Equivalent circuit Fig.. Transmission line model and its equivalent circuit This demonstrates that the power transfer capability is limited by voltage stability concern when the line length is greater than 88.7km. B. Extended Line Loadability Curve In this subsection, we will use the results obtained above to estimate the power transmission capabilities of 7k, 8k, k, k and k lines for different line length. The lines used for this study are shown in Tab. I. In this table, Z surge stands for surge impedance of the line. It can be seen from the table that the per-unit, per-km X and B values of overhead lines are all equal to.pu/km regardless of the voltage ratings. oltage TABLE I TYICAL LINE DATA AND UGE IMEDANCE LOADING (Ω/km) X (Ω /km) B (mω /km) Z surge (Ω) IL (MW) 7k k k k k.8.. er-unit vary.. The line loading limitations considered here are: thermal limitation, voltage stability limitation, voltage quality limitation, and angular stability limitation. A voltage drop of % is used as the voltage quality threshold. The angular stability limit is defined as the maximum transfer capability of the system. It should be noticed that in the orginal t. Clair curve [], the load angle (the corresponding stability margin is %) is selected as the angular stability limit. In this paper, in order to consist with the voltage stability limit, the load angle 9 is defined as angular stability limit.

3 Fig. and show the power transfer limits as a function of line length. The power level is expressed in per-unit of the IL of the respective lines. Fig. shows the limits when line charging is included while Fig. shows the results when the charging is compensated to zero. ower Transfer Limit (pu of IL) Thermal limit oltage stability limit oltage quality limit 8 Fig.. Loadability curves of transmission line (Uncompensated) ower Transfer Limit (pu of IL) oltage stability limit Thermal limit oltage quality limit 8 Fig.. Loadability curves of transmission line (Compensated) As discussed early, the per-unit X and B data are the same regardless of the line types. o each limit curve is applicable to all lines except the curve corresponding to the thermal limit. The thermal limit is line-dependent. A further note is that the lines are assumed to be lossless. This assumption is not quite accurate when dealing with low voltage lines. The impact of on the curves will be discussed in ection III. The results lead to the following conclusions: A short line is limited by thermal constraint. In the figures, the thermal limit curves slope down because when load increases, the increase of line current is not linear. The voltage quality limit is the most restrictive one as the line length increases. This is because voltage instability often occurs after the receiving end voltage drops beyond power quality limit. However, when the line length approaches to km, the voltage quality limit no longer exists. This is due to the voltage rise effect of long lines. If the line is not compensated, it is not possible to transfer power without causing voltage stability problem when the line length is over 88km. o the power transfer limit becomes zero from the voltage stability perspective. The cause of this phenomenon is explained in Fig. and (). The angular stability limit is the least restrictive one. However, the limit goes below. IL after the line approaches about 7km. If the line charging is compensated (Fig. ), it becomes possible to transfer some power over long distance without violating the voltage stability or voltage quality limits. However, the amount of power transferred is below. IL. C. Comparison of stability limits The practical stability limits of power systems are the load angle for the angular stability limit and the margin of % for the voltage stability limit. Fig. 7 and Fig. 8 show the power transfer capability curves with these realistic constrains. Fig.7 shows the limits when line charging is included while Fig. 8 shows the results when the charging is compensated to zero. ower Transfer Limit (pu of IL) oltage stability limit 8 Fig. 7. Comparison of realistic stability limits (Uncompensated) ower Transfer limit (pu of IL) oltage stability limit 8 Fig. 8. Comparison of realistic stability limits (Compensated) It can be observed from the figures that the angular stability limit is less restrictive than voltage stability limit under the above assumption. In Fig. 7, the angular stability limit reaches. IL when the line length is about 7km. This is slightly different from the original t. Clair curve in which the angular stability limit approaches. IL at 8km []. This difference is due to the fact that, in the original t. Clair curve, the combined reactance of step-up transformers and generators as well as of receiving systems was added directly

4 to the reactance of the line, while in this paper only the line reactance is considered. III. EFFECT OF YTEM AAMETE ON LINE LOADABILITY The effect of line resistance and shunt compensation are investigated at, 7, 8,,, and k transmission levels under the criteria of voltage drop of % and voltage stability constraint. A. Effect of Line esistance Fig. 9 shows the loadability curves with only voltage stability as concern for different transmission lines. It can be seen that high resistance (e.g. k line) will severely depress line loadability, particularly for short lines. This effect becomes much smaller for high voltage levels (e.g. k line). oltage stability limit (pu of IL) oltage rating increase =. (All voltages) =.78(k) =.8(7k) =.7(8k) =.9(k) =.(k) =.(k) 7 Fig. 9. oltage stability constrained power transfer limits (Uncompensated) Fig. and Fig. compare voltage stability limit and voltage quality limit for different transmission lines. In the figure, and Q stand for the voltage stability limit and voltage quality limit, respectively. oltage stability & quality limit =.78(k-) =.8(7k-) =.7(8k-) =.78(k-Q) =.8(7k-Q) =.7(8k-Q) 7 Fig.. oltage stability & quality limits with resistances for k, 7k, and 8 k lines (Uncompensated) It can be seen that for the given voltage levels, the voltage stability limit is higher than voltage quality limit, i.e., the voltage quality limit is more restrictive. B. Effect of hunt Compensation Compensating line charging is a common practice for high voltage transmission lines. This subsection further investigates the effect of shunt compensation on the power transfer capabilities. oltage stability & quality limit =.9(k-) =.(k-) =.9(k-Q) =.(k-q) 7 8 Fig.. oltage stability & quality limits with resistances for k and k lines (Uncompensated) Fig. and Fig. show the loadability curves with only voltage stability as concern for different transmission lines. In the figure, Compensated denotes that the line charging is fully compensated while Uncompensated denotes that the line is uncompensated. Fig. shows the power transfer limits for, 7, and 8 k lines while Fig. shows the power transfer limits for k and k lines. ower transfer limit (pu in IL) =.78(k-Compensated) =.8(7k-Compensated) =.7(8k-Compensated) =.78(k-Uncompensated) =.8(7k-Uncompensated) =.7(8k-Uncompensated) 7 Fig.. oltage stability constrained power transfer limits for k, 7k, and 8 k lines ower transfer limit (pu in IL) =.9(k-Compensated) =.(k-compensated) =.9(k-Uncompensated) =.(k-uncompensated) 7 8 Fig.. oltage stability constrained power transfer limits for k and k lines It can be seen that for low voltage levels (e.g. k), the loadability curve of the line with shunt compensation is much close to the curve without compensation. For high voltage

5 levels (e.g. k), when the line charging is not compensated, the line can not transfer power over certain distance (e.g. km) without violating the voltage stability constraints; however, when the line charging is compensated, it is possible to transfer some power over longer distance. Therefore, the effect of shunt compensation is to extend the line length for which the loadability is constrained by voltage stability. Fig. and Fig. compare voltage stability limit and voltage quality limit for different transmission lines which are fully compensated. Again, and Q stand for the voltage stability limit and voltage quality limit, respectively. Fig. shows the power transfer limits for, 7, and 8 k lines while Fig. shows the power transfer limits for k and k lines. It can be observed that the voltage quality limit is more restrictive. oltage stability & quality limit (pu) =.78(k-) =.8(7k-) =.7(8k-) =.78(k-Q) =.8(7k-Q) =.7(8k-Q) 7 uncompensated line, it is not possible to transfer power without causing voltage stability problem when the line length is over 88km.. The resistance has remarkable effect on line loadability, especially for low voltage levels. This effect will be decreased as voltage class increases.. With shunt compensation, it becomes possible to transfer power over long distance without violating the voltage stability or voltage quality limits.. EFEENCE [] H.. t. Clair, ractical concepts in capability and performance of transmission lines, A.I.E.E. Transactions art III. ower Apparatus and ystems, ol. 7, No., pp. -7, Jan. 9. []. Kundur, ower ystem tability and Control, McGraw-Hill, 99 [] J. D. Glover, M.. arma, ower ystem Analysis and Design, ( rd ed.) Brooks/Cole. Fig.. oltage stability & quality limits with resistances for k, 7k, and 8 k lines (Compensated) oltage stability & quality limit (pu) =.9(k-) =.(k-) =.9(k-Q) =.(k-q) Fig.. oltage stability & quality limits with resistances for k and k lines (Compensated) I. CONCLUION The universal t. Clair curve provides a means of depicting transmission line loadability as a function of its length. This paper further investigates the influence of voltage stability limit on the line loadability. tudies on the effects of various limiting factors lead to the following main conclusions:. The voltage quality limit has dominating influence on the loadability of short lines, while the voltage stability limit is the main constrain for long lines.. Both analytical and numerical results show that, for the