Lossless Distributed Parameter Lines

Reactive-Power Control in Electrical Power Transmission Systems

2.2 UNCOMPENSATED TRANSMISSION LINES .1 A Simple Case

2.2.2 Lossless Distributed Parameter Lines

Most power-transmission lines are characterized by distributed parameters:

series resistance,r; series inductance,l; shunt conductance,g; and shunt capac- itance,c—all per-unit (pu) length. These parameters all depend on the conduc- tors’ size, spacing, clearance above the ground, and frequency and temperature of operation. In addition, these parameters depend on the bundling arrangement of the line conductors and the nearness to other parallel lines.

The characteristic behavior of a transmission line is dominated by its l and cparameters. Parametersrandgaccount for the transmission losses. The fun- damental equations governing the propagation of energy along a line are the following wave equations:

d²V

d x² czyV (2.4a)

d²I

d x² czyI (2.4b)

20 REACTIVE-POWER CONTROL IN ELECTRICAL POWER TRANSMISSION SYSTEMS

where zyc(r+jql)(g+jqc).

For a lossless line, the general solutions are given as

V(x)cV_scosbx−j Z₀I_ssin bx (2.5a) I(x)cI_scosbx−j V_s

Z0

sin bx (2.5b)

These equations are used to calculate voltage and current anywhere on line, at a distance x from the sending end, in terms of the sending-end voltage and current and the line parameters. In Eqs. (2.4) and (2.5),

Z0c hl

c Q cthe surge impedance or characteristic impedance bcq

f

lcrad

/

^kmcthe wave number bacq

f

lcaradcthe electrical length of ana-km line

where l is the line inductance in henries per kilometer (H

/

^{km), c} is the line- shunt capacitance in farads per kilometer (F

/

^{km), and 1}

/

^flcis the propagation velocity of electromagnetic effects on the transmission line. (It is less than the velocity of light.)

From Eq. (2.5), we get

I_s c V_scosba−V_r j Z0sinba

If V_s cV_s/–0⁸ andV_r cV_r/– −d cV_r(cosd − jsind), then I_s c Vrsin d+j(Vrcosd−Vscosba)

Z₀sin ba (2.6)

Therefore, the power at the sending end is given as Ss cPs+jQs cVsI*

s

c V_sV_rsin d

Z0sinba +j V²_scosba−V_sV_rcosd

Z0sin ba (2.7)

Likewise, power at the receiving end is given as

UNCOMPENSATED TRANSMISSION LINES 21

x

aa / 2

aa V_s Z₀, b V_m

I_s I_r

P_s + j Q_s P_r + j Q_r

V_r

Figure 2.4 The power on a lossless distributed line.

SrcPr+jQr c− VsVrsind

Z₀sin ba +j V²_rcosba−VsVrcosd

Z₀sinba (2.8)

Comparing Eqs. (2.7) and (2.8) and taking the directional notation of Fig. 2.4 into account, it is concluded that for a lossless line, P_s c −P_r, as expected.

However,Q_s ⬆Q_r because of the reactive-power absorption

/

generation in the line.

From Eqs. (2.7) and (2.8), the power flow from the sending end to the receiv- ing end is expressed as

Pc V_sV_rsin d Z0sinba

In electrically short power lines, wherebais very small, it is possible to make a simplifying assumption that sin bac ba or Z₀sin bac Z₀ba c qla, where qlacX_l is the total series reactance of a line. This substitution results in the following well-recognized power equation:

Pc VsVr

X_l sind (2.9)

Accordingly, the maximum power transfer is seen to depend on the line length.

When the power-transfer requirement for a given length of a line increases, higher transmission voltages of Vs andVr must be selected.

This chapter is not intended to provide a comprehensive analysis of trans- mission lines. Rather, its objective is to examine those aspects that enhance the understanding of the interplay between voltages on the line and the resulting reactive-power flows.

2.2.2.1 Symmetrical Lines When the voltage magnitudes at the two ends of a line are equal, that is, V_s c V_r c V, the line is said to be symmetrical.

Because power networks operate as voltage sources, attempts are made to hold

22 REACTIVE-POWER CONTROL IN ELECTRICAL POWER TRANSMISSION SYSTEMS

almost all node voltages at nearly rated values. A symmetrical line, therefore, presents a realistic situation. From Eqs. (2.7) and (2.8) the following relation- ships are derived:

Ps c−Pr c V²

Z₀sinba sin d (2.10)

and

Qs cQr c V²cosba−V²cosd

Z0sin ba (2.11)

Active and reactive powers of a transmission line are frequently normalized by choosing the surge-impedance load (SIL) as the base. The SIL is defined as P0 cV²_nom

/

^{Z0, where Vnom} is the rated voltage. WhenVs cVr cVnom,

P_s P0 c−P_r

P0 c sind

sin bl (2.12)

and

Q_s Q0 c Q_r

Q0 c cosba

sinba − cos d

sin ba (2.13)

2.2.2.2 Midpoint Conditions of a Symmetrical Line The magnitude of the midpoint voltage depends on the power transfer. This voltage influences the line insulation and therefore needs to be well understood. For a symmetrical line where the end voltages are held at nominal values, the midpoint voltage shows the highest magnitude variation. In terms of the midpoint voltage Vm, the receiving-end voltage of a symmetrical line, from Eq. (2.4), is given as

Vr cVmcos ba

2 −j Z0Imsin ba

2 (2.14)

For simplification, defineV_mcV_m/–0⁸as the reference phasor. Because the line is symmetrical and lossless, that is,P_sc−P_r cP_mcPandQ_mc0, the midpoint current I_m is given by I_m cP

/

^Vm. Under these conditions, Eq. (2.14) can be rewritten as

V_r cV_mcos ba

2 −j Z₀ P Vm

sin ba 2 or

UNCOMPENSATED TRANSMISSION LINES 23

V²_r cV²_mcos² ba

2 +Z²₀ P²

V²_m sin² ba 2 Setting V_r cV_nomandV²_nom

/

^{Z0 cP0, we get}

V²_r

V²_nom c冢^VVnomm 冣^{2cos2 ba2 +}冢^VZ2nom⁰ 冣^2P2冢^VVnomm 冣^{2sin2 ba2}

If we letVm

/

^VnomcV˜m(per-unit voltage at the midpoint), then considerng that (Vr

/

^{Vnom)c 1, we have}

V˜⁴_m− _V˜²

m

cos² ba 2

+冢^PP0冣^{2tan2 ba2 c0}

Therefore

V˜ c



_





1 2cos² ba

2

± VU UU

T 1

4cos² ba 2

−冢^PP0 冣^{2tan2 ba2}

^

^



1/²

(2.15)

Equation (2.15) determines the midpoint voltage of a symmetrical line as a function of the power flowPon it.

Practical Considerations In general, the values of line parameters l and c remain reasonably independent of the transmission voltage. For example, typi- cal values ofl andcmay lie in the following ranges:

l cthe line inductance

/

^{km c}0.78–0.98 mH

/

^km

c cthe line capacitance

/

^kmc12.1–15.3 nF

/

^km

On the basis of these parameters, the surge impedance,Z0 c^fl

/

c, lies in the range of 225 to 285.

2.2.2.3 Case Study To illustrate a number of important considerations, let us choose a 735-kV symmetrical lossless transmission line with l c 0.932 mH

/

^{km, c c 12.2 nF}

/

km, and a line length of 800 km. From the foregoing parameters,

24 REACTIVE-POWER CONTROL IN ELECTRICAL POWER TRANSMISSION SYSTEMS V_s

P Q_r

V_r ∠−d

Q_c

P_l + jQ_l

Figure 2.5 The reactive-power balance at the receiving end.

Z₀ c hl

c c

h0.932

12.2 10³ c276.4Q Therefore, the SIL is

P₀c V²_nom

Z₀ c (735×10³)²

276.4 c1954.5MW

For this line to operate as a symmetrical line, that is, V_s c V_r c 735 kV, we have from Eq. (2.10):

P_sc V²

Z₀sin ba sind c 735²

276.4sin[(q^flc) 800]

sind

c2298.5sin dMWc1.176P₀sin dMW (2.16) It is important to calculate the required additional reactive power to hold the receiving-end voltage to 1 pu (735 kV). Let us assume that connected at the receiving end is a load of fixed power factor 0.9 lagging. For any load condition, the reactive-power balance at the receiving-end bus shown in Fig. 2.5 isQc c Qr +Ql, whereQr is the reactive-power flow from the receiving end into the line, Ql is the reactive-power component of the load, and Qc is the reactive power needed from the system to hold Vr to the rated value (1 pu).

Figure 2.6 shows Qr

/

^{P0, Ql}

/

^{P0 andQc}

/

^P0 as functions ofP

/

^P0. It should be observed that at no load (Pc0), nearly 1090-MVAR or 0.557-pu reactive power must be absorbed to hold the receiving-end voltage to 1 pu. To avoid overinsulating the line so that it might withstand overvoltages under no-load or light conditions, a common practice is to permanently connect shunt reactors at both ends to allow line energization from either end. Unfortunately, this natural protection becomes a liability under increased load conditions, for extra reactive power,Q_c, is needed to hold the terminal bus voltages at the desired level. The midpoint voltage of this line is calculated using Eq. (2.15), and typical voltage distribution on a distributed line is shown in Fig. 2.7.

Alternatively, consider the receiving half of the line. From Eq. (2.7), the power flow on the line is given as

UNCOMPENSATED TRANSMISSION LINES 25

0 0.2 0.2 0.4

−0.2

−0.4

−0.6 0.6 0.8 1 1.2

0.4 0.6 0.8

P /P₀ (pu)

Q (pu)

1.2 1

P₀ Q_c

P₀ Q_l

P₀ Q_r

Figure 2.6 The reactive-power flows at the receiving end.

P0 c V_mV_r Z₀sin ba

2

sin d

2 c V²_nom

Z₀sinba sin d

Because V_r cV_s cV_nom,

V˜_mc Vm

V_nom

sin ba 2 sinba

sin d sin d 2 c0.5724 sind

sin d 2

(2.17)

0.6 0.3 1 1.3

Receiving End Sending End

P /P₀= 0

1

V (pu)

Figure 2.7 The typical voltage distribution on a distributed line.

26 REACTIVE-POWER CONTROL IN ELECTRICAL POWER TRANSMISSION SYSTEMS

0 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15

0.2 0.4 0.6 P/P₀ (pu)

Load Angle, d (deg.)

Vm (pu)~

Vm~

d

0.8 1 1.20

10 20 30 40 50 60 70 80 90

Figure 2.8 The midpoint voltage,V, and load angle,˜ d, as functions ofP

/

^P0.

Figure 2.8 shows the load angledand the midpoint-normalized per-unit volt- age (_V˜_m) as functions ofP

/

^P0per-unit power transfer on the line. For this line, it is observed that for light loading below the surge-impedance load, the midpoint voltage exceedsVnomand reaches its highest value at no load. Furthermore, for stability considerations should an operating load angle of, say, 30⁸ be chosen to define the full-load rating of the line (0.588P0), the midpoint voltage of the line will be 1.1058 pu. These expected overvoltages in the range 0.1–0.2 pu from full load to no load are not within acceptable limits. Therefore, special techniques must be used to control these overvoltages.

It is possible to control the overall voltage profile of such a line as the one described in this case study by creating a midpoint voltage bus and connecting a controllable reactive-power source, called a var compensator, to it so that below the surge-impedance loading P₀, the var compensator absorbs reactive power, and aboveP₀, it supplies reactive power.

Figure 2.9 shows a midpoint var compensator employed as a voltage con- troller and the expected voltage profile along the line. Of course, it is not impor- tant to hold the midpoint voltageV_mc at 1-pu voltage, especially if there is no load connected to it. Also, it is not necessary to have a controllable var source at the midpoint; instead, an adequately sized fixed- or switched-shunt reactor could be used to keep the overvoltage within limits.

To continue this discussion with the aid of the case study, let us hold the midpoint voltage toVmc under all load conditions by employing a continuous var controller of unlimited capacity. Using Eq. (2.7), we have

Qmc V²_mccos ba

2 −VsVmccos d 2 Z₀sin ba

2

(2.18)

UNCOMPENSATED TRANSMISSION LINES 27 V_s= 1∠0^˚

Q_n V_mc ∠−d/2

Q_m Q_m

V_r= 1∠−d

Fixed Reactor or Continuous Var Controller

(a) P = 0

P < P₀ P > P₀ P = P₀

(b) 1

Figure 2.9 The midpoint-overvoltage control:Vmc c1pu.

Therefore, the var requirement from the midpoint var controller is

Qvc2Q_m (2.19)

In terms of the midpoint voltage, we can rewrite the power transfer of the line given by Eq. (2.7) as

P_compc VsVmc

Z₀sin ba 2

sin d

2 (2.20)

For the 735-kV, 800-km line with b c 1.27 × 10⁻³ rad

/

km, and assuming Vs cVr c1pu andVmc c1.05 pu, we get

b a

2 cb×400c0.508rad P_compc 735²×1.05

276.4×sin(0.508) sin d 2 c4215.28sin d

2 c2.157P₀sin d 2

28 REACTIVE-POWER CONTROL IN ELECTRICAL POWER TRANSMISSION SYSTEMS

0 20 500 1000 1149.25 1500 2000 2107.64

SIL = 1954.5

P_s P_comp

Q_u 2500

3000 3500 4000 4500

0 1000 433.54

−696.93 2000 3000 4000 5000 6000 7000 8000

30 40 60 80

Load Angle, d (deg.)

P (MW) Q (MVAR)

100 120 140 160 180

Figure 2.10 The relationship between active power, P, and reactive power,Q, with load angledin the 735-kV midpoint-compensated line.

Q_vc2Q_m c2× (735×1.05)²cos(0.508)−735²×1.05cos d 2 276.4sin(0.508)

c7740.98−8437.91cos d

2 c冢^{3.96−4.32cos d2}冣 ^P0

Figure 2.10 depicts the following relations:

Ps c2298.5 sin d MWcthe uncompensated line

P_comp c 4215.28 sin (d

/

^{2) MW c} the unlimited midpoint compensation to holdVmc c1.05 pu

Qvc7740.98 − 8437.91 cos (d

/

^{2) MVARc}the reactive power injected by the compensator SIL of the line

P₀ c1954 . 5MW

The results of Fig. 2.10 need careful interpretation. Note the following:

1. If the nominal rating of a line might correspond tod c30⁸, the 735-kV, 800-km symmetric line in this case study will be rated atP_nomc2298.5 sin(30⁸)c1149.25 MW, which is only 58.8% of the SIL.

UNCOMPENSATED TRANSMISSION LINES 29 2. If the preceding 735-kV symmetric line is provided with unlimited mid- point reactive-power compensation,Qv, to hold the midpoint voltage,Vmc, at 1.05 pu, then the maximum transferable power increases from 2298.5 MW (Ps in the uncompensated case) to 4215.28 MW (Pcompin the com- pensated case), implying a±83.39% increase. The nominal rating for this line could be 2107.64 MW (or 108% of SIL) fordc60⁸.

3. To maintain the midpoint voltage, V_mc, of the 735-kV symmetric line at 1.05 pu, the midpoint compensator’s var supply would range from

−696.93 MVAR to 7740.98 MVAR. This is a very large operating range for a line with 1954.5 MW SIL.

Therefore, let us search for a workable solution.

1. Assume that the midpoint-compensated line is rated atdc60⁸for stable operation, that is,Pcomp c2107.64 MW, with a nominal rating (1.08P0).

2. The midpoint-compensator power for dc60⁸is Qvc433.54 MVAR.

3. On the basis of entries 1 and 2, select a realistic midpoint var compensator rated to operate from −600to +400 MVAR.

The performance of the 735-kV, 800-km symmetric line, with a midpoint var compensator designed to operate at 1.05 pu terminal voltage in a controllable range, can be analyzed as follows: Beyond the limitQv>400MVAR, the var compensator behaves like a fixed capacitor of rating

Xcc V²_m

Qv c (1.05×735)²

400 c1489Q

In the uncontrollable range, from Eq. (2.18) the corresponding value of Qm in each half of the line is

Qm c V²_m

2Xc c V²_mcos ba

2 −V_sV_mcos d 2 Z₀sin ba

2 or

30 REACTIVE-POWER CONTROL IN ELECTRICAL POWER TRANSMISSION SYSTEMS

V_mc V_scos d 2 cos ba

2 − Z₀ 2Xc

sin ba 2

c 735cos d

2 cos(0.508)− 276.4

2×1489sin(0.508) c886.778cos d

2 c1.2065Vnomcos d 2

From this value of V_m (beyond the controllable capacitor range), the power flow on the line is given as

Pc VsVmsin d 2 Z₀sin ba

2

c 735×886.778sin d 2cos d

2 276.4sin(0.508) c2423.9sin d

The start of uncontrollable range on the capacitive side of the var compensator corresponds toQ_v>400MVAR at a value ofd calculated from

Qvc400c7740.98−8437.91cos d 2 or

dc59⁸

Likewise, belowQv<−600MVAR, the inductive limit of the var compensator is reached at a value ofd calculated from

Q_vc400c7740.98−8437.91cos d 2 or

dc17.38⁸

UNCOMPENSATED TRANSMISSION LINES 31 Thus

Pc4215.28sin(8.69⁸)c637.1MW

The value of the fixed inductor below dc17.38⁸ is calculated as X_lc V²_m

Qv c (735×1.05)²

600 c992.66Q since

Qmc −V²_m

2Xl c V²_mcos ba

2 −VsVmcos d 2 Z₀sin ba

2 or

V_mc Vscos d 2 cos ba

2 − Z₀ 2Xc

sin ba 2

c 735cos d

2 cos(0.508) + 276.4

2×992.66sin(0.508) c780.72cos d

2 c1.0622Vnomcos d 2

Using the foregoing value of V_m, power in the inductive uncontrollable range of the midpoint var compensator is given as

Pc VsVmsin d 2 Z₀sin ba

2

c 735×780.72sin d 2cos d

2 276.4sin(0.508) c2134sin d

32 REACTIVE-POWER CONTROL IN ELECTRICAL POWER TRANSMISSION SYSTEMS

0 20 500 1000 1500 2000 2500 3000 3500 4000 4500

40 60 80 100 120 140 160 180

0.2

P (MW)

0.4 0.6 0.8 1

Load angle d, (deg.)

17.38 59

P_s P P_comp

V_nom V_m

1.0622

Vm/Vnom (pu)

Figure 2.11 The relationship between powerP, midpoint voltageVm, and load angle d of a 735-kV symmetric line compensated by a−600

/

+400 MVAR range–controlled var compensator.

Figure 2.11 depicts following relations:

P_s c2298.5 sin d MWcthe uncompensated line

P_comp c 4215.28 sin (d

/

^{2) MW c} the unlimited midpoint compensation to holdV_mc c1.05 pu

Vm c1.0622Vnom cos(d

/

2) pu, for 0<d ≤17.38⁸ Pc2134sin d MW

V_m c1.05V_nom pu, for 17.38⁸≤d≤59⁸ Pc4215.28 sin(d

/

^{2) MW}

V_m c1.2065V_nom cos(d

/

2) pu, for 59⁸≤d≤180⁸ Pc2423.9 sin d MW

Figure 2.11 shows the results of a symmetric 735-kV, 800-km lossless line where, at its midpoint, a controllable var compensator is installed that holds the midpoint voltage at 1.05-pu value. The var compensator has a fully controllable range of−600MVAR to +400 MVAR. Beyond the +400-MVAR limit, the com- pensator behaves like a fixed capacitor (X_c c 1489 Q); below −600MVAR, it acts like a fixed inductor (Xlc992.66Q). It should be observed that under these

PASSIVE-COMPENSATION 33 circumstances, the maximum power transfer of the line is modified to 2423.9 MW, and the maximum overvoltage of the midpoint is limited to 1.062 pu.

The conclusions are as follows:

1. As the length of the line increases on account of the line-charging capac- itances, the line experiences significant overvoltages at light-load condi- tions.

2. Overvoltages can be limited by using fixed- or switched-shunt reactors at the line ends as well as at intermediate buses where needed.

3. The application of midpoint or intermediate bus-voltage controllers (var compensators) enhances the power-transmission capacity of a long line.

4. In practical cases, var controllers are sized by carefully selecting their continuous-operating range to hold the connecting-bus voltage within an acceptable range of values in the normal line-loading range.

Active and Passive Var Control When fixed inductors and

/

or capacitors are employed to absorb or generate reactive power, they constitute passive control.

An active var control, on the other hand, is produced when its reactive power is changed irrespective of the terminal voltage to which the var controller is connected.