Characterizing the Steady-State Performance of AC Lines Through a Dimensionless Universal Model

This paper revisits the AC power line model. Based solely on three dimensionless parameters, a universal electrical model is first developed, which is valid for any kind of power line (overhead, underground/submarine or gas insulated), irrespective of its rated voltage and length. Then, by neglecting the shunt losses, compact expressions are obtained in terms of the two remaining parameters for all electrical magnitudes characterizing the steady-state behavior of any line, such as real and reactive power flows at both ends, losses, charging current, transmission capability, etc. As a result, universal parameterized curves are also provided, including the well-known PV and P- $\delta $ curves. Moreover, numerical ranges of defining parameters are obtained for a diversity of present or future line arrangements, rated voltages and lengths, which are then represented as clusters of points on the universal curves. Beyond educational purposes, the proposed formulation and the associated characteristic curves can be useful, at the planning or design stages, when a certain degree of parameter uncertainty is still present, in order to compare different designs, or when selecting the most suitable simplified model for a particular case study, before fully defining the line features (type of conductor, phase-to-phase clearance, type of configuration, insulation, etc.).


B sh
Shunt susceptance (S). (B sh pu : per-unit magnitude). cos ϕ power factor of the load connected to the receiving end. C Per-unit-length shunt capacitance (F/km). g Per-unit-length shunt conductance (S/km).  Normalized sending-end current corresponding to p r = p rn . I s RMS current phasor at the sending end (A). l Length (km). l c Critical length (km). L Per-unit-length series inductance (H/km). p l Normalized active losses. p r Normalized magnitude of P r pu . P r pu Per-unit receiving-end active power (pu). p r0 Normalized power at the receiving end when q l = 0. p rmax Value of p r for δ = δ max . p rn Normalized magnitude of P rn . Normalized magnitude of P s pu . p sc Value of p s for p r = q r = 0. P s pu Per-unit sending-end active power (pu). q l Normalized reactive balance. q r Normalized magnitude of Q r pu . q sc Value of q s for p r = q r = 0. Q r pu Per-unit receiving-end reactive power (pu).

Q rn
Reactive power at the receiving end when active power is P rn and V r = 0.95V s (MW). q s Normalized magnitude of Q s pu . Q s pu Per-unit sending-end reactive power (pu). r Per-unit-length series resistance ( /km).

I. INTRODUCTION
Starting from the single-phase representation of two-port networks (customarily represented by the ABCD matrix), different distributed-parameter models of three-phase symmetrical lines have found widespread use, such as the π-equivalent, -equivalent or T-equivalent circuits [1]. For a given frequency (ω = 2πf ), the steady-state behavior of a power line can be represented through an equivalent circuit which depends ultimately on four electrical parameters, namely the series resistance and reactance (r and x = ωL), and the shunt conductance and susceptance (g and b = ωC), all per unit length, along with the line length l (which is worth retaining as an additional parameter to duly analyze its role).
Parametric studies that aim to be as universal as possible should, therefore, analyze the impact of these 4+1 parameters on the electrical performance of any line.
Historically, the presence of hyperbolic functions in the expressions of the ABCD matrix, and the lack of digital computers, led to the use of approximate expressions, obtained by retaining the first two or three terms of their series expansion. Alternatively, a long line can be divided into a finite number of short segments, each characterized by a lumped-parameter model, in which all parameters are proportional to the respective segment length. But none of those approximations can get rid of the five required parameters.
Simplified models, which take advantage of the large differences in parameter values found in several particular cases of practical interest, are routinely used, such as the four-parameter model (shunt conductance neglected), which is acceptable for virtually any overhead line, the three-parameter model (shunt admittance neglected), for relatively short overhead lines, or the lossless model (series resistance and shunt conductance neglected), usually adopted for long overhead lines rated at ultra high voltages [1].
Those approximations have proven useful in explaining a variety of power line behaviors, preferably in a graphical form to more easily capture the impact of each parameter. Additional parameterizations are frequently found, by considering for instance the rated voltage or external parameters, such as the power factor (or reactive power) of the load served by the line.
This paper develops an accurate positive-sequence analytical model, valid for any kind of AC line, which is based only on three dimensionless parameters. Such universal model leads to a rather simple algebraic formulation, from which the customary parameterized graphical analysis can be easily derived. Then, the performance of a generic line (voltage profile, transmission capability, reactive power internal balance, power losses, etc.), is fully characterized with the help of the proposed model.
The paper is divided as follows: Section II briefly provides the background (the classical distributed-parameter π-circuit). Section III reviews previous parametric studies that have used this formulation in its different simplifying variants. Section IV presents the new parameters and the corresponding dimensionless formulation of the line power flows. Section V presents the main analytical and graphical results derived from the application of the model on all types of lines, assuming the shunt conductance to be negligible. Finally, Section VI presents the conclusions.

II. DISTRIBUTED-PARAMETER π -CIRCUIT
The single-phase equivalent π-circuit of a power line, obtained from the ABCD matrix [1], is depicted in Fig. 1, where the total series impedance (Z s ) and half shunt admittance (Y sh ) can be expressed as nonlinear functions of five parameters, including the four per-unit-length values (r, x, g and b) and the line length, l, as follows: where z = r + jx, y = g + jb and γ = √ zy = α + jβ, Z 0 = √ z/y are known, respectively, as the propagation constant and the characteristic impedance. In the lossless case (r = g = 0), γ is purely imaginary (α = 0) and Z 0 reduces to a resistance (R 0 ). When the line is loaded with a pure resistance equal to R 0 , the operating point is known as Surge Impedance Loading (SIL), which is frequently used as a reference. It can be shown that any lossless line serving the SIL is perfectly balanced in terms of reactive power, and features a flat voltage profile [1].
For the purposes of this paper, the series admittance rather than the series impedance will be used: Note the minus sign preceding B s , intentionally used so that all line parameters are positive (this change of sign is not needed in (2) because B sh > 0). Accordingly, the current-voltage relationships for the single-phase circuit of Fig. 1 are as follows, when voltages are represented in polar coordinates: Then, the real and reactive power flows can be readily obtained from their definition (P + jQ = V I * ). At this point, it is customary to adopt the per unit system, which has two main advantages: 1) all transformer ratios become 1:1 (i.e., all impedances/admittances are referred to a common framework); 2) single-phase and three-phase magnitudes are indistinguishable for balanced systems. When doing so, the power flows at the sending (input power) and receiving (output power) ends, in per-unit magnitudes, are given by: In addition to the four line parameters, each equation embeds four electrical variables (power, voltage magnitudes and phase difference) which, independently or combined, are to be considered in the parameterized curves.

III. BRIEF REVIEW OF PARAMETRIC STUDIES
Since the beginnings of power engineering, the parametric analysis of lines has helped to better understand their electrical performance. Without claiming to be exhaustive, we summarize here some of those studies, focusing especially on aspects like parameters involved, simplifying assumptions and analytical and graphical results.
In [2] the aim of the parametric analysis was the estimation of the loadability characteristic curve (steady-state load capability versus length) of HV overhead lines (34.5−330 kV), obtaining empirically derived curves that are based on a predefined steady-state stability margin. In [3], a more analytical approach is performed, obtaining similar curves under the assumptions of lossless line and constant γ value for all line designs. This way, the load capability (in per-unit of SIL) is not dependent on per-unit-length parameters, being only a function of the line length and terminal voltages. Assuming a constant load power factor, it is possible to obtain simple curves of line loadability versus length, within voltage drop and stability limits. Moreover, the analysis of [2] is extended in [3] to EHV/UHV transmission lines, highlighting the fact that the simplifying assumptions are even more justified at those voltage levels. If the analysis is aimed at assessing the effect of the line design (single/double circuit, three/ four-conductor bundle, etc.) on its loadability characteristic, then a curve for each design should be considered [4]. Recently, a general formulation of the loadability characteristic regions (thermal limit, voltage drop limit and stability limit) has been proposed [5].
Regarding high voltage cables (66-220 kV), simple and approximate analytical expressions for the critical length (l c ) are derived in [6] by neglecting the series impedance Z s . Graphical representations of I r /I n versus l/l c , where I n is the cable ampacity, are plotted for a particular 220 kV, 2000 MCM high pressure, oil-filled pipe-type cable. Application of the same curves to other cross sections yields an acceptable error at voltages equal to or higher than 132 kV. When dry-type cables with low dielectric losses are rather considered, G sh can be neglected for l < 150 km. In [7], the loadability and losses of compensated, three-core submarine cables with voltages between 132 and 400 kV are depicted versus the cable length, for a particular cross section of 1000 mm 2 . In [8], an in-depth analysis of the behavior of umbilical cables with rated voltages between 35 and 150 kV and cable cross sections between 95 and 400 mm 2 is performed. The receiving end voltage and power, as well as the sending end current, are depicted versus the cable length in order to highlight the limitation of power transfer due to the charging current.
We can conclude that the previous parametric studies have usually been devoted to analyzing a particular aspect of a certain type of line (e.g. long overhead line, high voltage underground cable, etc.), generally applying some modeling assumptions and representing relevant magnitudes (e.g., receiving active power or sending charging current versus length) for a particular voltage range and configuration. This work is not intended to enhance those specific studies (well established in the literature), but rather to develop a comprehensive analytical and graphical framework, based on the distributed parameter model, which is accurate enough for all types of lines. The proposed unified framework uses three dimensionless parameters, which can be reduced for practical purposes to two parameters only. As the values of those parameters lie in a predefined interval for each major category of line, it is possible to assess the performance of a certain line, real or hypothetical, even if its parameters are not precisely known, by resorting to typical or expected values.

IV. DIMENSIONLESS PARAMETERS AND POWER FLOWS
To begin with, the following reference power, known as the Static Stability Limit (SSL), or static transmission capacity, is introduced: Note that, as the sending voltage is normally very close to the rated voltage, the SSL is mainly determined by the series admittance (i.e., by the tower/conductor configuration and line length). The reader should be aware that, by neglecting the series resistance (only valid for UHV lines), the SSL is also defined in some classical textbooks as V 2 s pu /X s pu . While the reasons for the name of this reference power will be apparent later on, it is worth noting at this point that the SSL is generally much larger than the SIL, also used as an alternative reference power in the literature.
Then, the power flow relationships (4)- (7) are normalized by dividing both members of the equations by the SSL, leading to: where v r = V r pu /V s pu denotes the receiving voltage normalized with the sending voltage. The above normalized power flows constitute the basis of the universal model developed in this work. They are nonlinear functions of the normalized receiving voltage (v r ) and the phase angle difference between the two end buses (δ), along with three dimensionless parameters that will be introduced in the sequel.

A. COEFFICIENT OF SERIES LOSSES
This represents the relative size of the series conductance (resistance) with respect to the series susceptance (reactance): The value of κ s ranges, for any kind of realistic AC line, between 0.05 (UHV transmission level) and 1.5 (LV feeder).

B. COEFFICIENT OF SHUNT LOSSES
This is the ratio between the shunt and series conductance: The value of κ sh is generally negligible, except for UHV lines facing very special environmental conditions. If neglected, as is customarily done, then the active power line losses (p l ) are proportional to κ s :

C. COEFFICIENT OF REACTIVE POWER BALANCE
This ratio represents the relative size of the shunt susceptance with respect to the series susceptance: A discussion follows, to better understand its range of values and role in the reactive power balance. From (16), taking into account (1) and (2), a very accurate approximation (less than 0.3 % error for any kind of power line), can be derived as follows: The ratio sin βl/βl is always close to 1, whereas κ s ranges between 0.05 and 1.5. Therefore, from (17), the following limits apply in practice: By giving extreme values to b, x and l we obtain the range of possible values for τ : Note that the (quadratic) effect of the line length on the electrical behavior of the line is integrated in this parameter, which explains its unusually wide range of variation if it has to cover all power lines of practical interest. From (10) and (12) the reactive power balance of the line is given by where a positive (negative) value means that the line has a deficit (surplus) of reactive power. Equation (20) compactly shows the important role of τ (and hence the line length) in the reactive power balance of the line, which justifies its name.

V. APPLICATIONS OF THE UNIVERSAL LINE MODEL
In this section, the steady-state electrical behavior of any kind of line, including the line loadability, efficiency, reactive power requirements, charging current, Ferranti effect and PV/QV/PQ curves, will be fully characterized based on the coefficients κ s and τ . For this purpose, it will be assumed that κ sh = 0 (i.e., G sh G s ). Hence, for simplicity of notation, κ s will be named just κ in the sequel.
A total of 20 power lines (see Table 1), covering the three major types (overhead, underground/submarine and gas insulated), as well as a wide range of rated voltages (from 400 V to 1200 kV), have been considered to illustrate the analysis and annotate the resulting curves. For each line, the length l is selected from the top range of typical present or future values. Along with the parameters, Table 1 includes the values of the active power at the receiving end, P rn (stability-limited in cases O6, O7 and O8, thermally-limited for the rest), as well as the reactive power Q rn calculated so that V r = 0.95V s for the given active power flow. Note that the reactive power is negative for overhead lines (i.e., it is also injected, rather than drawn, at the receiving end) and positive for cables, reflecting deficit and surplus, respectively. For the stabilitylimited lines, it is assumed that the maximum allowed phase angle difference is δ max /2, where δ max denotes the value of δ corresponding to the theoretical steady-state transmission capacity, as discussed below.
The values of κ and τ , corresponding to the lines of Table 1, are shown in Table 2, where a general trend for the evolution of those parameters with type of line and voltage can be observed. For the three line types, the behavior of κ is similar, namely decreasing from higher voltages. This is explained from (13) and (1): Typically, the ratio r/x decreases for increasing voltages, regardless the type of line. Note that, despite this general trend, there are particular cases, such as U3, departing from this rule of thumb, owing to the influence of other factors, like different conductor clearances in the underground case.
Regarding the evolution of τ with voltage and type of line, it is worth recalling from (18) that τ is basically proportional to bxl 2 . For each type of line, the dependence with the square of length is the main reason behind the behavior of τ . For overhead lines, the increasing length with voltage of the cases of Table 1 explains the evolution of τ from O1 to O8. On the other hand, for the underground cables, the evolution of length against voltage is basically decreasing, so the behavior of τ is the same. Eventually, for GILs of Table 1 the length is again increasing with voltage, so τ will behave similarly.
For the same reason, when comparing lines of similar length, even of different voltages, the difference in τ will be due basically to the product bx. A good example is O4 and G4, both with the same length (150 km). From Table 2, the values of τ for O4 and G4 are 134 and 125, respectively, which are roughly in the same proportion as their product bx.

A. LINE LOADABILITY
Regarding the behavior of lines in terms of active/reactive power flows, assuming that the sending voltage is equal to the rated voltage, it is customary to separately consider two idealized boundary constraints at the receiving bus, namely: 1) the voltage is specified (PV bus) and 2) the reactive power consumption is specified (PQ bus). In practice, the actual operating condition will lie somewhere between those extreme cases, depending on the type of line. For instance, a line belonging to a strongly meshed transmission system may be closer to the PV case, whereas a radial distribution line will be better characterized by the PQ-bus assumption. Both cases are considered below when analyzing the loadability from the steady-state stability viewpoint.

1) RECEIVING PV BUS
The maximum active power that can reach the receiving end for a given voltage (PV bus) can be obtained from (11) as VOLUME 10, 2022 follows: This limit depends to a greater extent on κ and to a lesser extent on v r , as can be seen in Fig. 2, that shows the curves of p r (δ) for different values of κ, with v r = 1 pu (top), and for different values of v r with κ = 0.1 (bottom).
In the lossless case (κ = 0), p rmax = v r . Therefore, if the receiving voltage is maintained at the rated voltage (v r = 1), then the maximum active power that can reach the receiving bus is exactly the SSL (p rmax = 1). But the same maximum power could be achieved in the lossy cases provided that v r > 1. It is worth noting that for v r ≤ 1 all the curves p r − δ have a common point at δ = arccos v r , regardless of the value of κ. Needless to say, angles larger than δ max correspond to infeasible steady-state operating points. In practice, a tighter limit than the SSL value is customarily imposed, in an attempt to address transient stability issues (e.g., sudden loss of a reactive power compensator near the receiving end). For instance, δ = δ max /2 is assumed in Table 1 to determine the maximum active power of lines O6, O7 and O8 (δ max is 84 • for O6 and 88 • for O7 and O8), but more conservative values, such as δ = δ max /3, are frequently found in the literature.

2) RECEIVING PQ BUS
The p r (δ) curves can also be obtained by imposing a given load power factor (cos ϕ) at the receiving bus (PQ bus). Eliminating v r by combining (11)- (12): leads to an expression of p r , not shown due to its complexity, which is depicted in Figure 3 for representative values of κ and τ . In this case, the maximum transmitted active power is roughly half the maximum value obtained for the PV-bus case. Note that the parameter τ affects the curves only slightly, whereas the influence of κ is higher, much like in Figure 2 (top).

B. LINE EFFICIENCY
From (9) and (11) the universal line efficiency expression is given by .
(24) Figure 4 represents the efficiency versus κ for different values of δ and two values of v r , namely v r = 1 (solid line) and v r = 0.9 (dashed line). As expected, the losses are null (η = 1) for κ = 0. For δ > 0, it can be shown that, for any v r < 1, the respective η curve intersects with that for v r = 1 exactly when η = v r . This can be observed in the figure, where the efficiency curves for v r = 1 and v r = 0.9 intersect, for a given δ > 0, at η = 0.9.

C. REACTIVE POWER BALANCE
In this subsection, the line behavior regarding the generation/ consumption of reactive power is analyzed for the PV and PQ receiving end cases.

1) RECEIVING PV BUS
In the PV-bus case, for a given v r , the reactive power balance of the line depends on the values of τ and δ. For small values of τ and δ, the first term of (20) is comparable to the second one, resulting in a narrow range of reactive power consumption/generation. On the contrary, for larger values of τ and δ the first term of (20) is lower and the second term is higher, giving rise to a wider range of reactive power consumption/generation.    Table 1 (v r = 0.95 pu). Figures 5 (overhead lines) and 6 (rest of lines), where q l is depicted versus δ for the values of τ corresponding to the whole set of lines in Table 1, when v r = 0.95 pu (qualitatively similar results are obtained for v r ≈ 1).

This is better observed in
In Figure 5, the medium voltage overhead lines (O1, O2) consume/generate a small amount of reactive power in the whole loading range (for v r ≈ 1) due to the small values of τ and δ. On the contrary, for high voltage overhead lines (O3 to O8) both terms of (20) may become quite different, leading to a much wider range of q l values.
For the underground/submarine and GIL lines (Figure 6), typically shorter than overhead lines, the conclusion is similar to that of the medium voltage lines, O1 and O2. In these cases, τ is small (it ranges from 2.8 · 10 −6 to 0.0265), so the first term of (20) is large, but the values of δ are also low (second  Table 1 (v r = 0.95 pu).
term of (20) non negligible), leading to relatively small reactive power imbalances. Accordingly, the range of variation between the loaded and unloaded cases is quite narrow, owing to the small range of variation of δ. As expected, as shown in Figure 6, the value of q l is mostly negative (surplus of reactive power). Be aware that, owing to the normalization of magnitudes, the scales of q l in figures 5 (overhead lines) and 6 (GIL and underground lines) are quite different, so directly comparing the q l values from both figures can be misleading.

2) RECEIVING PQ BUS
In the PQ-bus case, v r can be eliminated from (20), by means of (23). This way, q l becomes a function of δ, the power factor at the receiving end and the parameters κ and τ . Assuming cos ϕ = 0.9, the functions q l (δ) are represented in Figure 7   for the same values of κ and τ as in Figure 2. In contrast to Figure 2, τ is more influential in this case than κ in relation to the overall reactive power of the line. For light loads, there is an excess of reactive power, which intensifies for increasing values of τ and alleviates for increasing line loadings.

D. ACTIVE POWER FLOW THAT BALANCES THE REACTIVE POWER REQUIREMENTS OF THE LINE
In Figures 5 and 6 the horizontal axis (q l = 0) corresponds to the case in which the reactive power consumption of the series inductance is offset by the reactive power supplied by the shunt capacitance. Let p r0 denote the real power delivered by the line when q l = 0. In the lossless case, this operating point is termed the ''surge impedance loading'' (SIL), which is a reference or representative point worth characterizing. Figure 8 represents p r0 against τ for different values of κ and v r = 0.95 (similar results are obtained for v r ≈ 1). The cases O1, O2, U1 and U2 of Table 1 are not shown, since q l = 0 leads to an infeasible case, with no real solution for p r0 .
The normalized loadability limits (p rn ), provided in Table 1, are also shown in Figure 8. For the overhead lines considered, p rn is always higher than p r0 , which means that external reactive power support will be needed to deliver such levels of active power. Note, however, that for longer lengths than those assumed in Table 1 (higher values of τ ) the loadability limit might become lower than p r0 , which is uneconomic.
On the other hand, for underground/submarine and GIL lines it is generally verified that p rn < p r0 , meaning that a surplus of reactive power will always exist, regardless of the line loading. A notable exception is the line G1, that behaves much like an overhead line in this regard. This opens up the possibility of designing GIL lines at other voltages and lengths with a view to keeping the reactive unbalance at low levels.
It is possible to obtain an approximate yet accurate enough analytic expression for p r0 (v r ) by equating (20) to zero, from which The above approximations are acceptable as long as v r remains close to 1 (the practical case). For example, for v r = 0.95 the error in the approximation is 0.1%. Replacing (25) and (26) into (11) yields Note that, for the particular value τ = 1 − v r , the second term in (27) vanishes and p r0 is not a function of κ (as shown by the common point for all curves in Figure 8).
In the lossless case (κ = 0), p r0 reduces to the well-known SIL concept, for which the following simplified expression is obtained in terms of normalized parameters:

E. CHARGING CURRENT
The charging current i sc of a line is the current at the sending end with no load at the receiving end (p r = q r = 0). In some cases (typically in long cables) this unavoidable current can be so large that the remaining ampacity gets drastically reduced. By equating (11) and (12) to zero, the expressions for the no-load voltage at the receiving end and the real, reactive and apparent powers at the sending end are obtained. A further simplification is made by taking into account that τ 2, leading to: i sc = s sc ≈ 2τ where p sc and q sc denote the active and reactive power at the sending end when p r = q r = 0. The maximum error of the above approximations for the heterogeneous set of lines included in Table 1 is less than 1.5 %, except for the UHV long line O8, which is 4 %. Figure 9 shows the curves of i sc versus τ for different values of κ. The i sc values for the lines of Table 1 are shown as filled circles. In the same figure, the values of the current i smax , corresponding to the maximum load p rn for each line in Table 1, are also depicted as empty circles. The figure clearly shows a higher dependency of i sc on τ than on κ (the curves are quite close to each other). In fact, all the points corresponding to the lines of Table 1 lie virtually on the curve for κ = 0. The figure is also interesting to ascertain at a glance the relative importance of the no-load current over the maximum current, by comparing each filled circle with the empty one located right above. For the cases of Table 1, some lines like U3 or O8 have a significant no-load current that must be assessed and ultimately compensated, as it is well known.  For underground and GIL cases, the expression for i sc can be further simplified by noting that τ ≈ 0: The validity of this approximation can be checked in Figure 9, where the linear function i sc = 2τ is also represented.

F. FERRANTI EFFECT
Under the same no-load conditions of the previous section (p r = q r = 0), taking into account that τ 2, the voltage at the receiving end can be obtained from  Table 1.
which reflects the well-known Ferranti effect (open-circuit receiving end voltage always higher than the sending end FIGURE 11. PV generic curves for cos ϕ = 0.9 and some points of Table 1. voltage), more noticeable for lines with higher values of τ (long overhead lines). The receiving end voltage is represented in Figure 10 as a function of τ for different values of κ. The voltages corresponding to the lines of Table 1 are also shown as circles.

G. PV AND QV CURVES
By resorting only to κ and τ it is possible to obtain nearly exact generalized PV (nose) curves, where the receiving (unregulated) bus voltage is represented versus p r for a given load power factor (cos ϕ). Eliminating δ from (11) and (12) yields, Figure 11 shows the PV curves (upper part of the nose) for representative values of κ and two values of τ : 0 (representative of O1, O2, U1, U2, U6, U7, G1, G2, G3) and 0.02 (U3, U4, U5, G4, G5), assuming an inductive power factor cos ϕ = 0.9 (the typical case of a radial line with non regulated voltage at the load end). Moreover, the figure shows the points of maximum active power P rn of Table 1 and cos ϕ = 0.9. For the lines O3-O8, the P rn values shown in Table 1 are infeasible for a load with cos ϕ = 0.9 inductive (in fact, the reactive power at the receiving end should be negative, i.e. injected, for the operating points in the Table to be feasible). The PV curves corresponding to those lines are separately shown in Figure 12, where the infeasible points of active power P rn and cos ϕ = 0.9 are also represented.
From (34), it can be easily verified that v r = 1 for p r = 0 and τ = 0, regardless of the value of κ.
The QV curves can also be obtained for different values of κ and τ . These curves represent the required reactive power to be injected (q r < 0) or drawn (q r > 0) at the receiving end versus v r for a given p r value. The required relationship can be obtained by eliminating δ from (11) and (12), yielding: q r = τ + v 2 r (1 + κ 2 (1 − v 2 r )) − p 2 r − 2κp r v 2 r − 1 (35) As an example, Figure 13 shows the QV curves for p r = 0. From (35) it is immediately concluded that q r = τ for FIGURE 12. PV curves with cos ϕ = 0.9 for the lines O3-O8 of Table 1 and their corresponding points. p r = 0 and v r = 1, which is confirmed in the figure. This provides another interpretation for τ : the amount of (normalized) reactive power that must be drawn from the line at the receiving end to avoid the Ferranti effect at no load.

H. PQ CURVES
Other typical curves related to lines are those representing q r versus p r for a given value of v r , for which (35) is still of application when v r is replaced by a constant value. For example, for v r = 1, (35) reduces to,

VI. CONCLUSION
This paper has reconsidered the distributed-parameter, steady-state model of AC power lines in an attempt to develop a unified yet simple parameterized framework, from which the behavior of any power line, regardless of its type (overhead, underground, GIL), rated voltage and length, can be algebraically and graphically characterized. It is shown that only two dimensionless parameters (κ and τ ) are involved in the proposed generalized model (assuming the shunt conductance is negligible). Normalized power flows at both line terminals are inferred, allowing universal parameterized curves to be obtained for any electrical magnitude of interest, such as line losses, efficiency, no-load receiving end voltage and charging current, surge impedance loading, steady-state stability limit or PV, QV and PQ curves. The use of those parameters, which remain bounded in a known interval for all lines of practical interest, provides a more comprehensive and integrated understanding of the major issues at stake, when compared with the customary approach that segments lines by type, rated voltage and length. In addition to educational or academic uses, the universal model proposed in this work can be useful for planners and operators. For planners because, once they determine that the transmission capacity of a given congested corridor should be expanded, the dimensionless model can help them select the optimal arrangement among the several feasible choices. For operators because, given a power flow solution, or the state provided by an estimator in the EMS, they could perform an accurate ex-post graphical analysis of the performance (losses, closeness to the collapse point, etc.) of the different lines in the network, regardless of their type.