ࡱ> y{xZ!`  bjbj\\ 2>>6&vvvvNNN8.O$RPn2QV">V>V>V}W]XLX($h YyW}WYY v>V>V;(cccY8>V>VcYccq >VQ 0hQNx]<i|>0nͰƿ>_jƿxƿX"Xc YYXXX bpXXXnYYYY&,d",vvv Line Models and SIL Introduction In these notes, I present different line models that are used, and I also make some comments on Examples 4.2 and 4.3 leading to discussion of surge impedance loading, and finally I give a hint for problem 4.21. Simplified models (Section 4.5) We recall two things. First, we have the so-called exact transmission line equations:  EMBED Equation.3  (1)  EMBED Equation.3  (2) Second, we may represent a transmission line using a -equivalent model, shown below, if we use Z and Y /2, where  EMBED Word.Picture.8  Fig. 1  EMBED Equation.3  (3)  EMBED Equation.3  (4) Note that the two are equivalent, i.e., use of the -equivalent transmission model with Z and Y is equivalent to using eqs. (1), (2). Question: When is it OK to use the -equivalent transmission model with Z and Y (instead of Z and Y )? (Recall Z=zl, Y=yl where l is line length). Let s look at eqs. (3), (4) in more detail. They tell us that Z H"Z and YH"Y when  EMBED Equation.3  (5)  EMBED Equation.3  (6) To see when this happens, let s look up in a good math table how to express sinh(x) and tanh(x) as a Taylor s series. I used [, p. 58-59] to find that:  EMBED Equation.3   EMBED Equation.3  Using these in eqs. (5) and (6), we get:  EMBED Equation.3  (7)  EMBED Equation.3  (8) For both of these equations, they become true as the higher-order terms in the numerator get small relative to the first term in the numerator. This happens for small |l|, which occurs for small line length l. So when |l| is small, it is quite reasonable to use Z =Z=zl and Y =Y=yl. Consider a lossless line, i.e., a line for which r=0 in z=r+jx and g=0 in y=g+jb. (Note that for inductive series elements and capacitive shunt elements, that x and b will be positive numbers when defined with positive signs in z and y). Then  EMBED Equation.3 For transmission lines, g=0 always. So  EMBED Equation.3  (9) In the lossless case, r=0, and we get  EMBED Equation.3  (10) For a lossless line (=j), it is possible to show that =j=j0.0000013/meter [, pg 211] is quite typical for most transmission lines. For a 100 mile-long line:  EMBED Equation.3  Then:  EMBED Equation.3   EMBED Equation.3  which shows that sinh(l)H"l tanh(l/2)H" l/2 as required. The loss of accuracy from the approximation in these cases can be seen from:  EMBED Equation.3   EMBED Equation.3  The text recommends that lines longer than 150 miles should use Long-Line model, but [ NOTEREF _Ref146674119 \h 2] recommends that lines longer than about 100 miles should use the Long-Line model. Lines below about 100 miles may use Z=zl and Y=yl. Doing so results in the Medium-Length model, sometimes also referred to as the nominal -equivalent model. A final model suggested by the text is the short-length model, for lines shorter than 50 miles. This is the same as the Medium-Length model except Y is neglected altogether. This makes sense from the point of view that the parallel-plate capacitor in this case, can be considered to have short length, and thus a small area of the plates. Surge impedance loading Recall our definition of characteristic impedance ZC as:  EMBED Equation.3  (11) Example 4.2 considers a transmission line terminated in its characteristic impedance, per Fig. 2 (the long-line model is used).  EMBED Word.Picture.8 Fig. 2 One result of the analysis in Example 4.2 is to show that the complex power gain (the ratio of the power flowing out of the line to the power flowing into the line) is given by:  EMBED Equation.3  (12) where  is the attenuation constant (=+j). This says that when a line is terminated in ZC, the complex power gain (actually loss) is purely real. The implication of this is that the line (when terminated in ZC), only affects the real power (decreases it) but does not affect the reactive power at all. Consider reactive power implication: ( Whatever reactive power flows out of the line (and into the load) also flows into the line. So a line terminated in ZC has a very special character with respect to reactive power: the amount of reactive power consumed by the series X is exactly compensated by the reactive power supplied by the shunt Y, for every inch of the line! Another result of Example 4.2 is:  EMBED Equation.3  (13) It is the case that  is always non-negative. This means that el>1 and 0<e-l<1.0, and therefore |V2|<|V1|. So when the line is terminated in ZC, |V2| will always be less than or equal to |V1|. But what if the line is lossless? As shown by eq. (10), we see that =0. Referring back to eq. (13), this means that:  EMBED Equation.3  (14) This is a remarkable thing. If we terminate a lossless line in ZC, the voltage profile along the line will be flat! In other words, the voltage along the line will everywhere have the same magnitude. So |V2|=|V1| for lossless line terminated in ZC. Note also that lossless implies  EMBED Equation.3 (15) So ZC is purely real for the lossless case. So for lossless line terminated in ZC, since ZC is purely real, then only real power is delivered to it, and since the line is lossless, this same real power is delivered from the source. Therefore:  EMBED Equation.3  (16) The characteristic impedance is clearly an important parameter. It is also commonly referred to in the industry as the surge impedance. And the power flowing into it, per eq. (16), is called the surge impedance loading, denoted by PSIL, i.e.,  EMBED Equation.3  (17) Equation (17) gives SIL in terms of per-phase power (and line-to-neutral voltage). The text also gives it in terms of 3-phase power (and line-to-line voltage) as:  EMBED Equation.3  (18) Surge impedance and SIL are given for typical overhead 60 Hz three-phase transmission lines in Table 1 [, ]. Table 1 Vrated (kV) EMBED Equation.3  (ohms) EMBED Equation.3  (MW)69366-40012-13115380*35138366-40547-52161380*69230365-395134-145345280-366325-425500233-294850-1075765254-2662200-230011002315238* Estimated Line limits (Section 4.9) Fig. 3 is a well-known conceptual curve that captures some attributes of transmission lines (note that the numbers are typical and limits for any particular line may vary). Note the vertical axis is given as a percentage of SIL, simply because SIL provides a convenient characteristic of a transmission line that captures an attribute related to its power handling capability as a function of its physical construction. However, SIL does not capture the influence of length on power handling capability.  Fig. 3 These attributes are: Power limit decreases with line length Short lines are limited mainly by thermal problems. Medium length lines tend to be limited by voltage-related problems. Very long lines tend to be limited by stability problems. Complex power expression This material combines Sec. 4.6, 4.8, 4.9. Consider the long transmission line of Fig. 4. The voltages at the ends are specified as:  EMBED Equation.3   EMBED Equation.3  (19) The series impedance is  EMBED Equation.3  (20)  EMBED Word.Picture.8 Fig. 4 The complex power transferred into the line from bus 1 is:  EMBED Equation.3  (21) But IZ can be expressed as:  EMBED Equation.3  (22) Substitution of (22) into (21) yields:  EMBED Equation.3  (23) Distributing the V1 through, we obtain:  EMBED Equation.3  (24) where 12= 1- 2. Eq (24) is the same as eq. 4.29, pg. 104 in the text except eq. (24) also includes the effect of the sending-end line-charging, as represented by the last term. Circle diagrams If we ignore the last term in eq. (24), which means assuming no charging capacitance, and if we assume Z =Z, i.e., for the  short-line model, eq. (24) becomes:  EMBED Equation.3  Using Z*=|Z|e-j"Z, we get  EMBED Equation.3  (24a) We may repeat this same process to obtain power from the other direction as  EMBED Equation.3  The power flowing into the receiving bus from the line is then just S21, or  EMBED Equation.3  (24b) We can then define the following:  EMBED Equation.3 which are all constants if we assume the voltages are fixed. Then eqs. (24a) and (24b) become:  EMBED Equation.3  (24c)  EMBED Equation.3  (24d) Eqs. (24c) and (24d) characterize sending end and receiving end complex power as a function of the angle between the buses, 12. C1 and C2 are the centers of circles having radii |B|=|V1||V2|/|Z|. Figure 4.7 illustrates circle diagrams. The text makes 6 observations about these that I encourage you to review, among which are: #5: Transmission is strengthened by increasing voltages (generator field windings) and decreasing reactance (series compensation) #6: Strong coupling between P-flow and 12 and between Q-flow and |V|. An alternative development We want simple expressions for P12 and Q12. Recall eq. (3) above, repeated here:  EMBED Equation.3  (3) In the lossless case, =j, so eq. (3) is  EMBED Equation.3  (  EMBED Equation.3  (25) Likewise, we can derive, in lossless case:  EMBED Equation.3  (26) Here, C is total line capacitance. Eq. (25), (26) are same as (4.46), (4.45) in text, p. 115, and we see that Z and Y are pure reactances and susceptances, respectively, i.e., Z=jX and Y=jB. Conjugating eq. (25) and (26), we get:  EMBED Equation.3  (27)  EMBED Equation.3  (28) Recall eq. (24):  EMBED Equation.3  (24) Substitution of eqs. (27), (28) into the expression for S12, eq. (24), we get:  EMBED Equation.3 (29) Bringing the j of the first term into the numerator and canceling the two negative signs of the second term results in:  EMBED Equation.3 (30) Real power The first and last terms of eq. (30) are purely imaginary and so do not affect real power. 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hCJUVaJh6CJ0aJ0h<CJ0aJ0hCJ0aJ0h hI5CJ0aJ0hI5CJ0aJ0h CJ0aJ0h^CJ0aJ0h bCJ0aJ0jh^CJ0UaJ0!jhph^CJ0EHUaJ0jF h^CJUVaJ!~AAD\<ln~ĩDF(*x$^`a$gdk$a$gdnF#Of this term, we see that the real part is the sin12 term. Since this is the only real part of eq. (30), it must be true that:  EMBED Equation.3  (32) It is generally the case that voltage magnitudes at either end of a line are not very different (and when the line is lossless and terminated in ZC, they are exactly the same  see eq. (14) above). If we assume that |V2|=|V1|, then:  EMBED Equation.3  (33) Recalling eq. (17), repeated here:  EMBED Equation.3  (17) we see that eq. (33) can be re-written as:  EMBED Equation.3  (34) Although approximate, eq. (34) is very useful for getting a  back-of-the-envelope sense of what voltages are required in order to accommodate a given power transfer level, as observed in problem 4.21 of the homework assignment. If you take a position as a transmission planner, you might find this expression handy. Problem 4.21: Transient instability problems can occur when the angular separation between voltage phasors at either end of a transmission line, denoted by 12=1-2 becomes large. So when does 12 become large? Note that in eq. (34) that PSIL and sinl are constants for an already-constructed transmission line.  EMBED Equation.3  (34) So 12 directly determines (or is directly determined by) P12. As 12 increases from 0 to 90, the real power transfer P12 gets larger. Or, more properly, we can see that as P12 gets larger, 12 increases from 0 to 90. So answer to question of When does 12 become large? is: when P12 becomes large. Problem 4.21 indicates that a practical limit for transmission lines on 12 is 45 (actually, this is probably too large, a 30 limit is better, but we will use 45). It also indicates that the desired transfer level on a 300 mile-long line is 500 MW. Question is: what voltage levels can we consider? This is a design (planning) problem. The problem also indicates =0.002/mile. Solving eq. (34) for PSIL yields:  EMBED Equation.3  This is 3-phase power. Referring to Table 1 above, we see we need voltage level of at least 345 kV to accommodate this. Reactive power Consider again eq. (30), repeated here:  EMBED Equation.3 (30) Now let s consider the imaginary part. In doing so, we need to remember that the middle term contributes a real part, but it also contributes an imaginary part, as indicated by eq. (31), repeated here:  EMBED Equation.3  (31) Substituting eq. (31) into eq. (30), we have:  EMBED Equation.3  (35) Now taking only the imaginary part of (35):  EMBED Equation.3  (36) Recall that the denominator of the first two terms is just |Z | in the lossless case (see eq. (25) above). Therefore, we can write:  EMBED Equation.3  (37) Now let s consider eq. (37) for the short line (and lossless) model. In this case, Z H"Z, and for lossless, |Z|=X CH"0. So eq. (37) becomes:  EMBED Equation.3  (38) This should be familiar expression to you from EE 303. Voltage instability (Section 4.7) We can also write the real power equation (32) for the short line model, as we have just done for the reactive power equation, using Z =Z and |Z|=X, to get:  EMBED Equation.3  (39) Section 4.7 applies eqs. (38) and (39) to consider how the receiving end voltage magnitude varies with load. Assuming the (normalized) sending-end voltage is 1.0, the text derives:  EMBED Equation.3  (40) where |V2| is the normalized voltage at the receiving end, PD is the receiving end real power load =tan, where pf=cos (note  here is different from our previous use of it). Equation (40) can be used to plot receiving end voltage |V2| as a function of receiving end demand PD.  EMBED Word.Picture.8  Fig. 4 Fig. 4 is similar to Fig. E4.9 in the text. This figure is a very basic figure for transmission planners and operators, who constantly worry about the implications of this figure. Why?     PAGE  PAGE 24 [] R. Burington,  Handbook of Mathematical Tables and Formulas, 5th Edition, McGraw-Hill. [] O. Elgerd,  Electric Energy Systems Theory: An Introduction, 2nd edition, McGraw-Hill, 1982. [] J. Glover and M. Sarma,  Power System Analysis and Design, PWS Publishers, 1987. []  Transmission Line Reference Book: 345 kV and Above, Electric Power Research Institute, 2nd Edition, 1987. 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"S 21 =C 2 +Be "j 12ObjInfodEquation Native e_1188792432FZNZNOle h FMicrosoft Equation 3.0 DS Equation Equation.39q>p Z'=Z C sinhl=ZsinhllCompObjifObjInfokEquation Native l_1188792804FZNZNOle oCompObjpfObjInforEquation Native sX FMicrosoft Equation 3.0 DS Equation Equation.39q>< Z'=Z C sinhjl=Z C e jl "e "jl 2=Z C cosl+jsinl"cosl+jsinl2 FMicrosoft Equation 3.0 DS Equation Equation.39q>?p@ 2Z=jZ C sinl_1188792828FZNZNOle yCompObjzfObjInfo|Equation Native }[_1188793292 FZNZNOle CompObj  f FMicrosoft Equation 3.0 DS Equation Equation.39q>^ Y'=jCtan(l/2)l/2 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native z_1188793158 FZNZNOle CompObjfObjInfoEquation Native m_1188793776@FZNZN>Q 2Z * ="jZ C sinl FMicrosoft Equation 3.0 DS Equation Equation.39q>p  Y' * =Ole CompObjfObjInfoEquation Native "jCtan(l/2)l/2 FMicrosoft Equation 3.0 DS Equation Equation.39q,\+; S 12 =V 1  2 2Z _1220437095FZNZNOle CompObjfObjInfoEquation Native x_1188793831"FZNZNOle CompObj f* "V 1 V 2 e j 12 2Z * +V 1  2 2Y/2() * FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo!Equation Native _1188794540$FOOOle >¡1 S 12 =V 1  2 "jZ C sinl"V 1 V 2 e j 12 "jZ C sinl"jCV 1  2 tan(l/2)lCompObj#%fObjInfo&Equation Native _1188794633)FOO FMicrosoft Equation 3.0 DS Equation Equation.39q>™ S 12 =jV 1  2 Z C sinl+V 1 V 2 e j 12 jZ C sinl"jCV 1  2 tan(l/2)l FMicrosoft Equation 3.0 DS Equation Equation.39q> V 1 V 2 e jOle CompObj(*fObjInfo+Equation Native   12 jZ C sinl=V 1 V 2 e j 12 e "j90 Z C sinl=V 1 V 2 e j( 12 "90) Z C sinl=V 1 V 2 cos( 12 "90)+jsin( 12 "90)()Z C sinl=V 1 V 2 sin 12 "jcos 12 ()Z C sinl_1188795354'6.FOOOle CompObj-/fObjInfo0 FMicrosoft Equation 3.0 DS Equation Equation.39q>¶ P 12 =V 1 V 2 sin 12 Z C sinlEquation Native _11887954963FOOOle CompObj24f FMicrosoft Equation 3.0 DS Equation Equation.39q>£ P 12 =V 1  2 sin 12 Z C sinl FMicrosoft Equation 3.0 DS EqObjInfo5Equation Native _11887972821;8FOOOle CompObj79fObjInfo:Equation Native _1188803143=FOOuation Equation.39q>|p P 12 =P SIL sin 12 sinl FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj<>fObjInfo?Equation Native >  P SIL =P 12 sinlsin 12 =500sin(0.002300)sin45=400MW FMicrosoft Equation 3.0 DS EqJ%lPdXW\H`;d,hU =+ )5?;73./1E4Y'7o{ɭZ:I[k|{uoߦiټbуũˣљԝןܣޥϤwdmt}yÙѭ姧ȫïx{ӳѡ͎ח᡻ꫴ̩ئɹȯǨ˫ЯնٹؿƳ̳ӻƼĵðƱ˸͹ϻϿ¸ȸ˻ pHYsp IDATx EA.75-B 3kuc@@D$@?N5O <QƓka}J"2ɷf*ܢD_ࡑxݝ){x55S~^<`x 6 F<,/l`"`36x!K ,yXYjCct||2"t+(-T92">u]y7=t9Et1t:maN]oe^Ô"$ǭygB~A/ӉO3Wna: 5O?ѼƘ%L"DE{/IH?Md}Re^eILlBg\Ä~ "f09Z៑.!b. 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