- Email: [email protected]
Effect of load characteristics on maximum power transfer limit for HV compensated transmission lines
Electrical Power and Energy Systems 26 (2004) 467–472 www.elsevier.com/locate/ijepes
Effect of load characteristics on maximum power transfer limit for HV compensated transmission lines M.M. EL-Metwally, A.A. EL-Emary*, M. EL-Azab Department of Electrical power and Machines, Faculty of Engineering, Cairo University, Cairo, Giza, Egypt Received 5 February 2003; revised 14 August 2003; accepted 4 December 2003
Abstract In this paper a new method for calculating the maximum power transfer limit of high voltage compensated transmission lines is made. The effect of load characteristics as function of voltage and frequency is taken into account. Two schemes using series and shunt compensation are studied and numerical results for maximum power transfer limit, critical angular separation and critical voltage are given. These schemes are applied to the existing AC 500 kV transmission line connected between High-Dam and Cairo in Egypt. q 2004 Elsevier Ltd. All rights reserved. Keywords: Load modeling; Compensation transmission line; Maximum power transfer limit
1. Introduction Earlier, based on practical consideration and experience, St Clair [1], has derived loadability of uncompensated transmission lines and has expressed it in pu of the surge impedance loading (SIL) as a function of line length. Dunlop et al. [2], have presented the analytical basis for St Clair loadability curves and have extended their use into the EHV and UHV levels. They have shown that at higher voltages the loadability limits depend not only on the transmission line itself but also on the strengths of the terminal systems. They have stand that for EHV and UHV transmission lines, the only practical limitations to the line loadability are provided from considerations of line voltage drop and steady state stability as the thermal capabilities is significant only for very short lines. The line loadability characteristic derived analytically by Dunlop et al. does not take into consideration the var reserves available in the system. Recently Kay et al. [3], have shown the importance of var reserve modeling in determining the line loadability limits. They have shown that the conventional steady state stability is applicable only to transmission lines with unlimited var supplies. With finite var supplies, the loadability limits is decided by voltage stability rather than by conventional steady state stability. They have not considered the effects of line resistance and * Corresponding author. 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2003.12.008
line shunt susceptance and the distributed nature of the line model in determining the loadability. To over come the above disadvantage, Indulkar et al. [4,5], presented a modification of algorithm developed by Kay et al. This modification involves the determination of the line loadability limit taking into consideration the long line model of transmission line and the power factor of the load. They have shown that the loadability of lines with connected series capacitors and with finite var veserves depends on the voltage stability of the system. In addition to the use of series capacitors to improve the loadability, shunt reactors are required for voltage control [6 – 10]. In this paper the algorithm developed by Indulkar et al., is modified. The modification involves the determination of the line loadability limit, taking into consideration the long line model of the transmission line and load modeling as function of voltage and frequency. The effects of load characteristics on the maximum power transfer limit according to critical values of series and shunt compensation are calculated. Two different schemes using series and shunt compensation have been considered. These schemes are the same as these discussed in Ref. [11].
2. Proposed method The proposed method to obtain the degrees of series and shunt compensation to achieve the maximum power
468
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
Nomenclature X1 Bc Xc b kse ksh A; B Vr Vs Pr ; Q r PTC Por ; Qor m; n
total series inductive reactance of the line total shunt capacitive susceptance of the line reactance of series capacitor susceptance of shunt reactor degree of series compensation degree of shunt compensation generalized line constants after compensation receiving end voltage sending end voltage active and reactive power loading of the line power transfer capability active and reactive power loading of the line at Vr ¼ 1.0 pu constant of voltage characteristic of the load
transfer limit Pr;crit: of a transmission line is summarized as follows:
a; g Ka ; Ta f J R k1 – k6 Vt Te 0 Tdo Efd d 0 Eq M; D Xd ; Xq
constant of frequency characteristic of the load exciter gain and time constant frequency Jacobean speed regulation in per unit constant of the linearized model of synchronous machine terminal voltage electromagnetic torque d-axis transient open circuit time constant exciter field voltage torque angle voltage proportional to direct axis flux linkages inertia constant and damping coefficient d-axis and q-axis reactance
by considering a set of nonlinear equations as follows: F1 ¼ Por Vrm f a 2
Vs Vr sin d B
ð5Þ
F2 ¼ Qor Vrn f g 2
Vs V r AVr2 cos d þ B B
ð6Þ
2.1. System equations In this paper two schemes of compensation are considered: (scheme i and scheme ii) as shown in Fig. 1. Assuming the line is lossles, the active and reactive power flows at the end of transmission line with series and shunt compensation are given by: Pr ¼
Vs Vr sin d B
ð1Þ
Qr ¼
Vs V r AVr2 cos d 2 B B
ð2Þ
F3 ¼ AVr þ Vs sin d tan f 2 Vs cos d
ð7Þ
F4 ¼ Vs cos d 2 AVr 2 ðf Þðg2aÞ Vrðn2mÞ Vs sin d tan f
ð8Þ
where F3 and F4 are derived in Appendix (A.1) It is shown that Eqs. (5), (6) and (8) are function of Vr ; f ; d; kse and ksh : Equations for calculating the relation between f and d are derived in Appendix (A.2) [13,14]. This relation is given by the following formula
where A and B are the generalized constants of the compensated transmission line.
f ¼ 2RH d
2.2. Load model
F1 ¼ Por Vrm ð2RH dÞa 2
Vs Vr sin d B
ð10Þ
F2 ¼ Qor Vrn ð2RH dÞg 2
Vs Vr AVr2 cos d þ B B
ð11Þ
The active and reactive load demand varies with voltage and frequency according the following functions Pr ¼ Por Vrm f a
ð3Þ
Qr ¼ Qor Vrn f g
ð4Þ
ð9Þ
Rewriting Eqs. (5) – (8) according to Eq. (9), yields
These functions are the same as these discussed in Ref. [17]. Also, the values a and g are given in this reference. Por and Qor are the nominal values of active and reactive power of load, respectively. 2.3. Mathematical development Active and reactive power in Eqs. (1) and (2) must also satisfy the load active and reactive power of Eqs. (3) and (4). For a given value of the degrees of series and shunt compensation, the maximum power transfer is determined
Fig. 1. Compensation schemes.
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
F3 ¼ AVr þ Vs sin d tan f 2 Vs cos d
ð12Þ
F4 ¼ Vs cos d 2 AVr 2 ð2RHÞðg2aÞ Vrðn2mÞ dðg2aÞ Vs sin d tan f ð13Þ The well known Newton’s method is used to the solution of Eqs. (10) –(13). The Newton’s method requires that a set of linear equation be formed as follow [15] ðkÞ 1 0 DF1 C B B DF C B 2C C¼ B C B B DF3 C A @ DF4
ðkÞ 0 1 B C BJC B C B C B C B C @ A
where J is given by. 0 › F1 › F 1 B ›d › Vr B B B › F2 › F 2 B B › Vr B ›d J¼B B ›F ›F 3 B 3 B B ›d › Vr B B @ › F4 › F 4 ›d › Vr
ðkÞ 1 0 Dd C B B DV C B r C C B C B B Dkse C A @
› F2 ›kse › F3 ›kse › F4 ›kse
3.1. The uncompensated line Without using any type of compensation the following results are obtained for one circuit of the line. Po ¼ 900 MW and PTC ¼ 1:325Po MW these results are based on Vs ¼ 500 kV and MVA ¼ 2100: To operate the line with a transmission angle of d ¼ 308; the power transfer Pt ¼ 0:6625Po MW. 3.2. Results of the effect of load characteristics on maximum power transfer limit
ð14Þ
Dksh
› F1 ›kse
469
1 ›F 1 ›ksh C C C ›F 2 C C C ›ksh C C ›F 3 C C C ›ksh C C C ›F 4 A
›ksh
The Jacobean matrix J is given in Appendix (A.3) The element of the Jacobean matrix and F1 to F4 in Eq. (14), are evaluated by substituting the current values of ðkÞ ðkÞ VrðkÞ ; dðkÞ ; kse and ksh : Hence a solution for the Dd; DVr ; Dkse and Dksh can be obtained by solving a system of linear ðkþ1Þ ðkþ1Þ equations. The new values for Vrðkþ1Þ ; dðkþ1Þ kse and ksh are calculated from:
dðkþ1Þ ¼ dðkÞ þ DdðkÞ Vrðkþ1Þ ¼ VrðkÞ þ DVrðkÞ ðkþ1Þ ðkÞ ðkÞ kse ¼ kse þ Dkse ðkþ1Þ ðkÞ ðkÞ ksh ¼ ksh þ Dksh
This process is repeated until to successive solutions differ only by a specified tolerance equal to 0.001, to get the degree of series and shunt compensation and the corresponding values of critical angle dcrit: critical voltage Vr crit: ; and critical power Pr crit: ; for different types of load models.
3. Application to a practical system To test the capability of the proposed method the application of series capacitor and shunt reactor compensation to the existing 500 kV transmission line in Egypt is made. This transmission line is connected between the High-Dam generating plant in Upper Egypt and Cairo. The system data is given in Appendix (A.4).
3.2.1. Effect of voltage dependent load characteristics The effect of voltage dependent load characteristics is obtained by setting g ¼ a ¼ 0:0: The results of maximum power transfer limit of the transmission line, degree of series and shunt compensation with the given data are determined. Both voltage dependent and voltage independent characteristics of the load are considered. For voltage dependent load, the loads characteristics parameters considered are m ¼ n ¼ 1:0; m ¼ n ¼ 2:0 [18], and m ¼ n ¼ 0:5: The results are summarized as follows: i
ii
iii
iv
Constant power load model. Table 1 shows the computational results at m ¼ n ¼ 0:0; corresponding to a loading of 0.5 – 2.5 SIL. Constant current load model. Table 2 shows the computational results for different line loading of Pr as function SIL at m ¼ n ¼ 1:0 Constant impedance load model. Table 3 shows the computational results for different line loading of Pr as function of SIL at m ¼ n ¼ 2:0: Composite load model. Table 4 shows the computational results for different line loading of Pr as function of SIL at m ¼ n ¼ 0:5 [4].
From Tables 1– 4, it can be observed that the critical angle dcrit: is independent of loading of the transmission line. The degrees of series and shunt compensation, critical power Pr crit: and Vr crit: are dependent of transmission line Table 1 Effect of constant power load model ðm ¼ n ¼ 0:0Þ Pr =SIL
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Pr crit: (pu)
a. Scheme i 0.5 30 1.0 63 2.1 72 2.0 77 2.5 80
73 75 78 80 82
0.31 0.31 0.31 0.31 0.31
0.9 0.86 0.84 0.84 0.83
0.725 1.45 2.18 2.91 3.6
b. Scheme ii 0.5 39 1.0 76 1.5 86 2.0 90 2.5 93
77 80 82 85 85
0.31 0.31 0.31 0.31 0.31
0.87 0.84 0.83 0.82 0.82
2.15 3.11 3.92 4.61 4.96
kse (%)
470
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
Table 2 Effect of constant current load model ðm ¼ n ¼ 1:0Þ Pr =SIL
kse (%)
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Table 5 Effect of load characteristics on line loadability limits. m ¼ n ¼ 1:0 and a ¼ g ¼ 20:3 Pr crit: (pu)
a. Scheme i 0.5 20 1.0 57 1.5 69 2.0 75 2.5 78
73 74 76 79 81
0.3 0.3 0.3 0.3 0.3
0.9 0.86 0.84 0.83 0.83
0.65 1.25 1.85 2.46 3.01
b. Scheme ii 0.5 27 1.0 69 1.5 82 2.0 88 2.5 92
77 79 81 83 85
0.31 0.31 0.31 0.31 0.31
0.88 0.84 0.83 0.83 0.82
2.06 2.86 3.60 4.2 4.6
Pr =SIL
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Pr crit: (pu)
a. Scheme i 0.5 70 1.0 84 1.5 86 2.0 87
79 81 89 89
0.436 0.436 0.436 0.436
0.74 0.73 0.72 0.72
1.87 3.73 5.62 7.5
b. Scheme ii 0.5 70 1.0 87 1.5 87 2.0 88
69 74 76 78
0.450 0.450 0.450 0.450
0.73 0.72 0.72 0.72
2.81 3.55 3.83 4.94
kse (%)
loading. Pr crit: ; kse and ksh increase by increase of loading. On otherhand Vr crit: decreases. It can be also seen that the critical voltage Vr crit: and critical power Pr crit of voltage independent load in Table 1 are greater that critical voltage Vr crit: and critical power Pr crit of voltage dependent load given in Tables 2– 4. Also, the results show that the critical angular is independent of line loading but is effected by load characteristics parameters m and n:
3.2.2. Effect of voltage and frequency dependent load Table 5 shows the effects of voltage sensitive and frequency sensitive load, for m ¼ n ¼ 1:0 and g ¼ a ¼ 20:3 [17]. It can be observed that the line loadabilty limits has been affected for the case of voltage sensitive and frequency sensitive more than voltage sensitive only. The critical power and critical angle are increased while critical voltage is decreases.
Table 3 Effect of constant Impedance load model ðm ¼ n ¼ 2:0Þ
4. Conclusions
Pr =SIL
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Pr crit: (pu)
a. Scheme i 0.5 15 1.0 52 2.1 65 2.0 72 2.5 75
72 74 75 77 78
0.31 0.31 0.31 0.31 0.31
0.91 0.86 0.85 0.84 0.84
0.607 1.09 1.59 2.09 2.58
b. Scheme ii 0.5 20 1.0 62 2.1 77 2.0 85 2.5 89
76 78 80 82 83
0.3 0.3 0.3 0.3 0.3
0.89 0.85 0.83 0.83 0.82
2.02 2.61 3.26 3.82 4.29
kse (%)
In this paper, a new method has been developed for assessing the loadability limit of HV compensated transmission lines taking into consideration the effect of load characteristics as function of voltage and frequency. The application of this method to 500 kV existing transmission line of Upper Egypt is made. It is noted that from this application the load characteristic plays an important role on maximum power transfer limit, critical angular separation and critical voltage. Appendix A A.1. Determination of function F3 and F4
Table 4 Effect of load characteristics ðm ¼ n ¼ 0:5Þ Pr =SIL
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Pr crit: (pu)
a. Scheme i 0.5 25 1.0 60 2.1 71 2.0 76 2.5 79
72 75 77 79 81
0.3 0.3 0.3 0.3 0.3
0.9 0.86 0.84 0.84 0.83
0.688 1.34 2.0 2.67 3.629
b. Scheme ii 0.5 33 1.0 73 2.1 84 2.0 89 2.5 90
77 78 82 84 87
0.3 0.3 0.3 0.3 0.3
0.87 0.84 0.84 0.82 0.82
2.1 3.0 3.78 4.45 4.87
kse (%)
A.1.1. Derivation of F3 The sending end voltage of transmission line with a given series and shunt compensation is given by [12]: Vs ¼ AVr þ BIr
ðA1Þ
Eq. (A1), is rewritten in the form: Pr 2 jQr Vrp
ðA2Þ
Vs Vr cos d þ jVs Vr sin d ¼ AVr2 þ BQr þ jBPr
ðA3Þ
Vs /d ¼ AVr þ jB
Equating the real and imaginary parts of Eq. (A3) gives: Vs Vr cos d ¼ AVr2 þ BQr
ðA4Þ
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
Vs Vr sin d ¼ BPr
ðA5Þ
From Eqs. (A10) to (A13) DVt ¼ {ðk5 2 k6 k3 k4 Þ=ð1 þ k6 k3 ka Þ}Dd
From Eqs. (A4) and (A5) F3 ¼ AVr þ Vs sin d tan f 2 Vs cos d
471
DTe ¼ {ðk1 2 k2 k3 k4 ÞDd 2 ðk2 k3 ka Þðk5 2 k6 k3 k4 Þ= ðA6Þ
ð1 þ k6 k3 ka Þ}Dd þ {ðk2 k3 ka Þð1 2 k6 k3 ka =1 þ k6 k3 ka Þ} ðA14Þ
A.2.2. Derivation of F4 Pr and Qr must be satisfy the load characteristics then: Qor ¼ Por tan f
ðA7Þ
ðA15Þ or DPe ¼ HDd
From Eqs. (3) and (4) P Qr ¼ mr a tan fVrn f g Vr f
DPe ¼{ðk1 2k2 k3 k4 Þ2ðk2 k3 ka Þðk5 2k6 k3 k4 Þ=ð1þk6 k3 ka Þ}Dd
ðA16Þ
where ðA8Þ H ¼ {ðk1 2 k2 k3 k4 Þ 2 ðk2 k3 ka Þðk5 2 k6 k3 k4 Þ=ð1 þ k6 k3 ka Þ}
or Qr ¼ Pr Vrðn2mÞ ð2RH dÞðg2aÞ tan f Substituting Eqs. (1) and (2) into Eq. (A8) gives. F4 ¼ Vs cos d 2 AVr 2 ð2RHÞðg2aÞ Vs Vrðn2mÞ ðdÞðg2aÞ sin d tanf ðA9Þ
A.2.2. Governor model Generator real power output is adjusted by the static response of prime mover as shown in Fig. A2. Real power output may be expressed as: Pe ¼ 2 R1 f
ðA17Þ
From Eqs. (A16) and (A17) HDd ¼ 2 R1 Df
A.2. Relation between frequency and power angle d
or
The relation between frequency f and power angle d is obtained as follows: A.2.1. Generator model The change in terminal voltage, electrical power, field voltage and proportional voltage are given by (Fig. A1): 0
DVt ¼ k5 Dd þ k6 DEq 0
ðA10Þ
DTe ¼ k1 Dd þ k2 DEq
ðA11Þ
DEfd ¼ ka {u 2 DVt þ DVref: }
ðA12Þ
0
DEq ¼ k3 DEfd 2 k3 k4 Dd
ðA13Þ
f ¼ 2RH d
ðA18Þ
A.3. Element of Jacobean matrix J Assuming the line is lossless the generalized constants A and B for compensation model shown in Fig. 1, (scheme ii as an example) are given below. A ¼ ð1 2 1=2kse ksh X1 Bc Þcos u 1 1 X 2 X1 B2c Zo kse ksh 2 2 Bc Zo ksh 2 1 kse sin u; 8 2 2Zo 1 1 X2 2 2 2 ksh 2 1 kse B ¼ j Zo 2 kse ksh X1 Bc Zo þ X12 B2c Zo Kse 2 16 4Zo
1 2 2 X B k k 2 X1 kse cos u sin u þ 4 1 c se sh The element of Jacobian matrix is given as follows:
aPr Vs V r cos d 2 d B mPr V ›F1 =›Vr ¼ 2 s sin d Vr B ›F1 =›d ¼
Fig. A1. Shown the block diagram of generator model [13,14].
Fig. A2. Governor model.
472
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
b ›F1 =›kse ¼ Vs Vr sin d 12 : B b ›F1 =›ksh ¼ Vs Vr sin d 22 B gQr Vs Vr sin d ›F2 =›d ¼ þ d B nQr V ›F2 =›Vr ¼ 2 s cos d Vr B b a B2b A ›F2 =›kse ¼ Vs Vr cos d 12 þ Vr2 1 2 1 B B b a B2b A ›F2 =›ksh ¼ Vs Vr cos d 22 þ Vr2 2 2 2 B B
›F3 =›d ¼ Vs cos d tan f þ Vs sin d ›F3 =›Vr ¼ A ›F3 =›kse ¼ a1 Vr ›F3 =›ksh ¼ a2 Vr ›F4 =›d ¼ 2 Vs sin d 2 ð2RHÞðg2aÞ Vs tan fVrðn2mÞ ½cos ddðg2aÞ þ ðg 2 aÞdðg2a21Þ
›F4 =›Vr ¼ 2 ½A þ ðn 2 mÞð2RHÞðg2aÞ Vs tan f sin ddðg2aÞ Vrðn2m21Þ
›F4 =›kse ¼ 2a1 Vr ›F4 =›ksh ¼ 2a2 Vr where a1 ; a2 ; b1 and b2 are given as follows: 1 1 X1 2 2 a1 ¼ cos u 2 ksho X1 Bc 2 sin u X1 Bc Zo ksho 2 2 8 2Zo 1 1 1 a2 ¼ cos u 2 kseo X1 Bc 2 sin u X1 B2c Zo kseo ksho 2 Bc Zo 2 4 2 ! 1 1 2 2 X12 2 k b1 ¼sin u 2 ksho X1 Bc Zo þ X1 Bc Zo kseo ksho 2 2 8 2Zo se 1 þ cos u X12 Bc kseo ksho 2 X1 2 1 1 2 ksho b2 ¼ sin u 2 kseo X1 Bc Zo þ X12 B2c Zo kseo 2 8 1 2 þ cos u X12 Bc Ksco 4 A.4. System and load data The 500 kV High Dam-Cairo transmission system has the following data [10,16]. Vs ¼ 500 kV MVA ¼ 2100 MW Line reactance, X1 ¼ 0:3020 V/km Line susceptance, Bc ¼ 3:9 £ 1026 s/km
Line length, L ¼ 788 km Generator: 0 0 Xd ¼ 0:275 pu; Xq ¼ 0:275 pu; Xd ¼ 0:05 pu; Tdo ¼ 2:8 s: M ¼ 29:0 Excitation: Ka ¼ 50 and Ta ¼ 0:02 s: Initial operation point: P ¼ 1:0 pu Q ¼ 0:4 pu Vt ¼ 1:0 pu
References [1] St Clair HP. Practical concepts in capability and performance of transmission lines. AIEE transmission (power apparatus and system) paper 53-338 presented at the AIEE specific general meeting, Vancouver, BC, Canada; September 1–4 1953. [2] Dunlop RD, Gutman R, Marchenko PP. Analytical development of loadability characteristics for EHV and UHV transmissions lines. IEEE, Trans PAS-98 1979;606–17. [3] Kay TW, Sauer PW, Smith RA. EHV and UHV line loadability dependence on VAR supply. IEEE Trans Power Apparatus Syst 1982; PAS-101(9):3568–75. [4] Indulkar CS, Viswanathan B, Venkata SS. Reactive Power Constrained Loadability Limits Of Series Capacitor Compensation EHV transmission lines. IEEE Trans Power Syst 1987;2(3):585 –91. [5] Indulkar CS, Viswanathan B, Venkata SS. Maximum power transfer limited by voltage stability in series and shunt compensated schemes for AC transmission systems. IEEE Trans Power Deliv 1989;4(2):1246–52. [6] Miller TJE. Reactive power control in electric systems. New York: Wiley; 1982. [7] Ashok Kumar BS, Parthasarathy K, Prabhakara FS, Khincha HB. Effectiveness of series capacitor in long distance transmission lines. IEEE Trans PAS-89 1970;941–51. [8] Rajaramain R, Alvarado F, Maniaci A, Camfield R, Jalali S. Determination location and amount of series compensation to increase power transfer capability. IEEE Trans Power Syst 1998;13(2):294–300. [9] Woodford DA, Tranawecky MZ. Compensation of long distance AC transmission lines by shunt connected reactance controllers. IEEE, Trans PAS-94 1975;655–64. [10] EL-Marsafawy M. Application of series capacitor and shunt reactor compensation to an existing practical AC transmission line. IEE ProcC 1991;138(4):330–6. [11] Iliceto F, Cinieri E. Comparative analysis of series and shunt compensation schemes for AC transmission system. IEEE Trans PAS-96 1977;1819–30. [12] Steveson WD. Element of power system analysis. New York: McGraw-Hill; 1982. [13] Yu YN. Electric power system dynamic. India: Academic Press Inc; 1983. [14] EL-Emary A, EL-Arini M, EL-Alaily A. Fast contingency evaluation method for generator outage simulation and power system security assessment. ETEP 1995;5(2):129–32. [15] Stagg GW, El-Abiad AH. Computer methods in power system analysis. New York: McGraw-Hill; 1968. [16] FRCU 830202. University Linkage Project, Cairo University. Dynamic equivalent models and decentralized multi level control of unified power system of Egypt. Report 2; 1984. [17] Okamura M, O-ura Y, Hayashi S, Uemura K, Ishiguro F. A new power flow model and solution method including load and generator characteristics and effects of system control devices. IEEE Trans Power Apparatus Syst 1975;PAS-94(3):1042–50. [18] IEEE Task Force on Load Representation for Dynamic Performance, Load representation for dynamic performance analysis. IEEE Trans Power Syst 1993;8(2):472– 82.
Effect of load characteristics on maximum power transfer limit for HV compensated transmission lines M.M. EL-Metwally, A.A. EL-Emary*, M. EL-Azab Department of Electrical power and Machines, Faculty of Engineering, Cairo University, Cairo, Giza, Egypt Received 5 February 2003; revised 14 August 2003; accepted 4 December 2003
Abstract In this paper a new method for calculating the maximum power transfer limit of high voltage compensated transmission lines is made. The effect of load characteristics as function of voltage and frequency is taken into account. Two schemes using series and shunt compensation are studied and numerical results for maximum power transfer limit, critical angular separation and critical voltage are given. These schemes are applied to the existing AC 500 kV transmission line connected between High-Dam and Cairo in Egypt. q 2004 Elsevier Ltd. All rights reserved. Keywords: Load modeling; Compensation transmission line; Maximum power transfer limit
1. Introduction Earlier, based on practical consideration and experience, St Clair [1], has derived loadability of uncompensated transmission lines and has expressed it in pu of the surge impedance loading (SIL) as a function of line length. Dunlop et al. [2], have presented the analytical basis for St Clair loadability curves and have extended their use into the EHV and UHV levels. They have shown that at higher voltages the loadability limits depend not only on the transmission line itself but also on the strengths of the terminal systems. They have stand that for EHV and UHV transmission lines, the only practical limitations to the line loadability are provided from considerations of line voltage drop and steady state stability as the thermal capabilities is significant only for very short lines. The line loadability characteristic derived analytically by Dunlop et al. does not take into consideration the var reserves available in the system. Recently Kay et al. [3], have shown the importance of var reserve modeling in determining the line loadability limits. They have shown that the conventional steady state stability is applicable only to transmission lines with unlimited var supplies. With finite var supplies, the loadability limits is decided by voltage stability rather than by conventional steady state stability. They have not considered the effects of line resistance and * Corresponding author. 0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2003.12.008
line shunt susceptance and the distributed nature of the line model in determining the loadability. To over come the above disadvantage, Indulkar et al. [4,5], presented a modification of algorithm developed by Kay et al. This modification involves the determination of the line loadability limit taking into consideration the long line model of transmission line and the power factor of the load. They have shown that the loadability of lines with connected series capacitors and with finite var veserves depends on the voltage stability of the system. In addition to the use of series capacitors to improve the loadability, shunt reactors are required for voltage control [6 – 10]. In this paper the algorithm developed by Indulkar et al., is modified. The modification involves the determination of the line loadability limit, taking into consideration the long line model of the transmission line and load modeling as function of voltage and frequency. The effects of load characteristics on the maximum power transfer limit according to critical values of series and shunt compensation are calculated. Two different schemes using series and shunt compensation have been considered. These schemes are the same as these discussed in Ref. [11].
2. Proposed method The proposed method to obtain the degrees of series and shunt compensation to achieve the maximum power
468
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
Nomenclature X1 Bc Xc b kse ksh A; B Vr Vs Pr ; Q r PTC Por ; Qor m; n
total series inductive reactance of the line total shunt capacitive susceptance of the line reactance of series capacitor susceptance of shunt reactor degree of series compensation degree of shunt compensation generalized line constants after compensation receiving end voltage sending end voltage active and reactive power loading of the line power transfer capability active and reactive power loading of the line at Vr ¼ 1.0 pu constant of voltage characteristic of the load
transfer limit Pr;crit: of a transmission line is summarized as follows:
a; g Ka ; Ta f J R k1 – k6 Vt Te 0 Tdo Efd d 0 Eq M; D Xd ; Xq
constant of frequency characteristic of the load exciter gain and time constant frequency Jacobean speed regulation in per unit constant of the linearized model of synchronous machine terminal voltage electromagnetic torque d-axis transient open circuit time constant exciter field voltage torque angle voltage proportional to direct axis flux linkages inertia constant and damping coefficient d-axis and q-axis reactance
by considering a set of nonlinear equations as follows: F1 ¼ Por Vrm f a 2
Vs Vr sin d B
ð5Þ
F2 ¼ Qor Vrn f g 2
Vs V r AVr2 cos d þ B B
ð6Þ
2.1. System equations In this paper two schemes of compensation are considered: (scheme i and scheme ii) as shown in Fig. 1. Assuming the line is lossles, the active and reactive power flows at the end of transmission line with series and shunt compensation are given by: Pr ¼
Vs Vr sin d B
ð1Þ
Qr ¼
Vs V r AVr2 cos d 2 B B
ð2Þ
F3 ¼ AVr þ Vs sin d tan f 2 Vs cos d
ð7Þ
F4 ¼ Vs cos d 2 AVr 2 ðf Þðg2aÞ Vrðn2mÞ Vs sin d tan f
ð8Þ
where F3 and F4 are derived in Appendix (A.1) It is shown that Eqs. (5), (6) and (8) are function of Vr ; f ; d; kse and ksh : Equations for calculating the relation between f and d are derived in Appendix (A.2) [13,14]. This relation is given by the following formula
where A and B are the generalized constants of the compensated transmission line.
f ¼ 2RH d
2.2. Load model
F1 ¼ Por Vrm ð2RH dÞa 2
Vs Vr sin d B
ð10Þ
F2 ¼ Qor Vrn ð2RH dÞg 2
Vs Vr AVr2 cos d þ B B
ð11Þ
The active and reactive load demand varies with voltage and frequency according the following functions Pr ¼ Por Vrm f a
ð3Þ
Qr ¼ Qor Vrn f g
ð4Þ
ð9Þ
Rewriting Eqs. (5) – (8) according to Eq. (9), yields
These functions are the same as these discussed in Ref. [17]. Also, the values a and g are given in this reference. Por and Qor are the nominal values of active and reactive power of load, respectively. 2.3. Mathematical development Active and reactive power in Eqs. (1) and (2) must also satisfy the load active and reactive power of Eqs. (3) and (4). For a given value of the degrees of series and shunt compensation, the maximum power transfer is determined
Fig. 1. Compensation schemes.
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
F3 ¼ AVr þ Vs sin d tan f 2 Vs cos d
ð12Þ
F4 ¼ Vs cos d 2 AVr 2 ð2RHÞðg2aÞ Vrðn2mÞ dðg2aÞ Vs sin d tan f ð13Þ The well known Newton’s method is used to the solution of Eqs. (10) –(13). The Newton’s method requires that a set of linear equation be formed as follow [15] ðkÞ 1 0 DF1 C B B DF C B 2C C¼ B C B B DF3 C A @ DF4
ðkÞ 0 1 B C BJC B C B C B C B C @ A
where J is given by. 0 › F1 › F 1 B ›d › Vr B B B › F2 › F 2 B B › Vr B ›d J¼B B ›F ›F 3 B 3 B B ›d › Vr B B @ › F4 › F 4 ›d › Vr
ðkÞ 1 0 Dd C B B DV C B r C C B C B B Dkse C A @
› F2 ›kse › F3 ›kse › F4 ›kse
3.1. The uncompensated line Without using any type of compensation the following results are obtained for one circuit of the line. Po ¼ 900 MW and PTC ¼ 1:325Po MW these results are based on Vs ¼ 500 kV and MVA ¼ 2100: To operate the line with a transmission angle of d ¼ 308; the power transfer Pt ¼ 0:6625Po MW. 3.2. Results of the effect of load characteristics on maximum power transfer limit
ð14Þ
Dksh
› F1 ›kse
469
1 ›F 1 ›ksh C C C ›F 2 C C C ›ksh C C ›F 3 C C C ›ksh C C C ›F 4 A
›ksh
The Jacobean matrix J is given in Appendix (A.3) The element of the Jacobean matrix and F1 to F4 in Eq. (14), are evaluated by substituting the current values of ðkÞ ðkÞ VrðkÞ ; dðkÞ ; kse and ksh : Hence a solution for the Dd; DVr ; Dkse and Dksh can be obtained by solving a system of linear ðkþ1Þ ðkþ1Þ equations. The new values for Vrðkþ1Þ ; dðkþ1Þ kse and ksh are calculated from:
dðkþ1Þ ¼ dðkÞ þ DdðkÞ Vrðkþ1Þ ¼ VrðkÞ þ DVrðkÞ ðkþ1Þ ðkÞ ðkÞ kse ¼ kse þ Dkse ðkþ1Þ ðkÞ ðkÞ ksh ¼ ksh þ Dksh
This process is repeated until to successive solutions differ only by a specified tolerance equal to 0.001, to get the degree of series and shunt compensation and the corresponding values of critical angle dcrit: critical voltage Vr crit: ; and critical power Pr crit: ; for different types of load models.
3. Application to a practical system To test the capability of the proposed method the application of series capacitor and shunt reactor compensation to the existing 500 kV transmission line in Egypt is made. This transmission line is connected between the High-Dam generating plant in Upper Egypt and Cairo. The system data is given in Appendix (A.4).
3.2.1. Effect of voltage dependent load characteristics The effect of voltage dependent load characteristics is obtained by setting g ¼ a ¼ 0:0: The results of maximum power transfer limit of the transmission line, degree of series and shunt compensation with the given data are determined. Both voltage dependent and voltage independent characteristics of the load are considered. For voltage dependent load, the loads characteristics parameters considered are m ¼ n ¼ 1:0; m ¼ n ¼ 2:0 [18], and m ¼ n ¼ 0:5: The results are summarized as follows: i
ii
iii
iv
Constant power load model. Table 1 shows the computational results at m ¼ n ¼ 0:0; corresponding to a loading of 0.5 – 2.5 SIL. Constant current load model. Table 2 shows the computational results for different line loading of Pr as function SIL at m ¼ n ¼ 1:0 Constant impedance load model. Table 3 shows the computational results for different line loading of Pr as function of SIL at m ¼ n ¼ 2:0: Composite load model. Table 4 shows the computational results for different line loading of Pr as function of SIL at m ¼ n ¼ 0:5 [4].
From Tables 1– 4, it can be observed that the critical angle dcrit: is independent of loading of the transmission line. The degrees of series and shunt compensation, critical power Pr crit: and Vr crit: are dependent of transmission line Table 1 Effect of constant power load model ðm ¼ n ¼ 0:0Þ Pr =SIL
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Pr crit: (pu)
a. Scheme i 0.5 30 1.0 63 2.1 72 2.0 77 2.5 80
73 75 78 80 82
0.31 0.31 0.31 0.31 0.31
0.9 0.86 0.84 0.84 0.83
0.725 1.45 2.18 2.91 3.6
b. Scheme ii 0.5 39 1.0 76 1.5 86 2.0 90 2.5 93
77 80 82 85 85
0.31 0.31 0.31 0.31 0.31
0.87 0.84 0.83 0.82 0.82
2.15 3.11 3.92 4.61 4.96
kse (%)
470
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
Table 2 Effect of constant current load model ðm ¼ n ¼ 1:0Þ Pr =SIL
kse (%)
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Table 5 Effect of load characteristics on line loadability limits. m ¼ n ¼ 1:0 and a ¼ g ¼ 20:3 Pr crit: (pu)
a. Scheme i 0.5 20 1.0 57 1.5 69 2.0 75 2.5 78
73 74 76 79 81
0.3 0.3 0.3 0.3 0.3
0.9 0.86 0.84 0.83 0.83
0.65 1.25 1.85 2.46 3.01
b. Scheme ii 0.5 27 1.0 69 1.5 82 2.0 88 2.5 92
77 79 81 83 85
0.31 0.31 0.31 0.31 0.31
0.88 0.84 0.83 0.83 0.82
2.06 2.86 3.60 4.2 4.6
Pr =SIL
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Pr crit: (pu)
a. Scheme i 0.5 70 1.0 84 1.5 86 2.0 87
79 81 89 89
0.436 0.436 0.436 0.436
0.74 0.73 0.72 0.72
1.87 3.73 5.62 7.5
b. Scheme ii 0.5 70 1.0 87 1.5 87 2.0 88
69 74 76 78
0.450 0.450 0.450 0.450
0.73 0.72 0.72 0.72
2.81 3.55 3.83 4.94
kse (%)
loading. Pr crit: ; kse and ksh increase by increase of loading. On otherhand Vr crit: decreases. It can be also seen that the critical voltage Vr crit: and critical power Pr crit of voltage independent load in Table 1 are greater that critical voltage Vr crit: and critical power Pr crit of voltage dependent load given in Tables 2– 4. Also, the results show that the critical angular is independent of line loading but is effected by load characteristics parameters m and n:
3.2.2. Effect of voltage and frequency dependent load Table 5 shows the effects of voltage sensitive and frequency sensitive load, for m ¼ n ¼ 1:0 and g ¼ a ¼ 20:3 [17]. It can be observed that the line loadabilty limits has been affected for the case of voltage sensitive and frequency sensitive more than voltage sensitive only. The critical power and critical angle are increased while critical voltage is decreases.
Table 3 Effect of constant Impedance load model ðm ¼ n ¼ 2:0Þ
4. Conclusions
Pr =SIL
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Pr crit: (pu)
a. Scheme i 0.5 15 1.0 52 2.1 65 2.0 72 2.5 75
72 74 75 77 78
0.31 0.31 0.31 0.31 0.31
0.91 0.86 0.85 0.84 0.84
0.607 1.09 1.59 2.09 2.58
b. Scheme ii 0.5 20 1.0 62 2.1 77 2.0 85 2.5 89
76 78 80 82 83
0.3 0.3 0.3 0.3 0.3
0.89 0.85 0.83 0.83 0.82
2.02 2.61 3.26 3.82 4.29
kse (%)
In this paper, a new method has been developed for assessing the loadability limit of HV compensated transmission lines taking into consideration the effect of load characteristics as function of voltage and frequency. The application of this method to 500 kV existing transmission line of Upper Egypt is made. It is noted that from this application the load characteristic plays an important role on maximum power transfer limit, critical angular separation and critical voltage. Appendix A A.1. Determination of function F3 and F4
Table 4 Effect of load characteristics ðm ¼ n ¼ 0:5Þ Pr =SIL
ksh (%)
dcrit: (rad.)
Vr crit: (pu)
Pr crit: (pu)
a. Scheme i 0.5 25 1.0 60 2.1 71 2.0 76 2.5 79
72 75 77 79 81
0.3 0.3 0.3 0.3 0.3
0.9 0.86 0.84 0.84 0.83
0.688 1.34 2.0 2.67 3.629
b. Scheme ii 0.5 33 1.0 73 2.1 84 2.0 89 2.5 90
77 78 82 84 87
0.3 0.3 0.3 0.3 0.3
0.87 0.84 0.84 0.82 0.82
2.1 3.0 3.78 4.45 4.87
kse (%)
A.1.1. Derivation of F3 The sending end voltage of transmission line with a given series and shunt compensation is given by [12]: Vs ¼ AVr þ BIr
ðA1Þ
Eq. (A1), is rewritten in the form: Pr 2 jQr Vrp
ðA2Þ
Vs Vr cos d þ jVs Vr sin d ¼ AVr2 þ BQr þ jBPr
ðA3Þ
Vs /d ¼ AVr þ jB
Equating the real and imaginary parts of Eq. (A3) gives: Vs Vr cos d ¼ AVr2 þ BQr
ðA4Þ
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
Vs Vr sin d ¼ BPr
ðA5Þ
From Eqs. (A10) to (A13) DVt ¼ {ðk5 2 k6 k3 k4 Þ=ð1 þ k6 k3 ka Þ}Dd
From Eqs. (A4) and (A5) F3 ¼ AVr þ Vs sin d tan f 2 Vs cos d
471
DTe ¼ {ðk1 2 k2 k3 k4 ÞDd 2 ðk2 k3 ka Þðk5 2 k6 k3 k4 Þ= ðA6Þ
ð1 þ k6 k3 ka Þ}Dd þ {ðk2 k3 ka Þð1 2 k6 k3 ka =1 þ k6 k3 ka Þ} ðA14Þ
A.2.2. Derivation of F4 Pr and Qr must be satisfy the load characteristics then: Qor ¼ Por tan f
ðA7Þ
ðA15Þ or DPe ¼ HDd
From Eqs. (3) and (4) P Qr ¼ mr a tan fVrn f g Vr f
DPe ¼{ðk1 2k2 k3 k4 Þ2ðk2 k3 ka Þðk5 2k6 k3 k4 Þ=ð1þk6 k3 ka Þ}Dd
ðA16Þ
where ðA8Þ H ¼ {ðk1 2 k2 k3 k4 Þ 2 ðk2 k3 ka Þðk5 2 k6 k3 k4 Þ=ð1 þ k6 k3 ka Þ}
or Qr ¼ Pr Vrðn2mÞ ð2RH dÞðg2aÞ tan f Substituting Eqs. (1) and (2) into Eq. (A8) gives. F4 ¼ Vs cos d 2 AVr 2 ð2RHÞðg2aÞ Vs Vrðn2mÞ ðdÞðg2aÞ sin d tanf ðA9Þ
A.2.2. Governor model Generator real power output is adjusted by the static response of prime mover as shown in Fig. A2. Real power output may be expressed as: Pe ¼ 2 R1 f
ðA17Þ
From Eqs. (A16) and (A17) HDd ¼ 2 R1 Df
A.2. Relation between frequency and power angle d
or
The relation between frequency f and power angle d is obtained as follows: A.2.1. Generator model The change in terminal voltage, electrical power, field voltage and proportional voltage are given by (Fig. A1): 0
DVt ¼ k5 Dd þ k6 DEq 0
ðA10Þ
DTe ¼ k1 Dd þ k2 DEq
ðA11Þ
DEfd ¼ ka {u 2 DVt þ DVref: }
ðA12Þ
0
DEq ¼ k3 DEfd 2 k3 k4 Dd
ðA13Þ
f ¼ 2RH d
ðA18Þ
A.3. Element of Jacobean matrix J Assuming the line is lossless the generalized constants A and B for compensation model shown in Fig. 1, (scheme ii as an example) are given below. A ¼ ð1 2 1=2kse ksh X1 Bc Þcos u 1 1 X 2 X1 B2c Zo kse ksh 2 2 Bc Zo ksh 2 1 kse sin u; 8 2 2Zo 1 1 X2 2 2 2 ksh 2 1 kse B ¼ j Zo 2 kse ksh X1 Bc Zo þ X12 B2c Zo Kse 2 16 4Zo
1 2 2 X B k k 2 X1 kse cos u sin u þ 4 1 c se sh The element of Jacobian matrix is given as follows:
aPr Vs V r cos d 2 d B mPr V ›F1 =›Vr ¼ 2 s sin d Vr B ›F1 =›d ¼
Fig. A1. Shown the block diagram of generator model [13,14].
Fig. A2. Governor model.
472
M.M. EL-Metwally et al. / Electrical Power and Energy Systems 26 (2004) 467–472
b ›F1 =›kse ¼ Vs Vr sin d 12 : B b ›F1 =›ksh ¼ Vs Vr sin d 22 B gQr Vs Vr sin d ›F2 =›d ¼ þ d B nQr V ›F2 =›Vr ¼ 2 s cos d Vr B b a B2b A ›F2 =›kse ¼ Vs Vr cos d 12 þ Vr2 1 2 1 B B b a B2b A ›F2 =›ksh ¼ Vs Vr cos d 22 þ Vr2 2 2 2 B B
›F3 =›d ¼ Vs cos d tan f þ Vs sin d ›F3 =›Vr ¼ A ›F3 =›kse ¼ a1 Vr ›F3 =›ksh ¼ a2 Vr ›F4 =›d ¼ 2 Vs sin d 2 ð2RHÞðg2aÞ Vs tan fVrðn2mÞ ½cos ddðg2aÞ þ ðg 2 aÞdðg2a21Þ
›F4 =›Vr ¼ 2 ½A þ ðn 2 mÞð2RHÞðg2aÞ Vs tan f sin ddðg2aÞ Vrðn2m21Þ
›F4 =›kse ¼ 2a1 Vr ›F4 =›ksh ¼ 2a2 Vr where a1 ; a2 ; b1 and b2 are given as follows: 1 1 X1 2 2 a1 ¼ cos u 2 ksho X1 Bc 2 sin u X1 Bc Zo ksho 2 2 8 2Zo 1 1 1 a2 ¼ cos u 2 kseo X1 Bc 2 sin u X1 B2c Zo kseo ksho 2 Bc Zo 2 4 2 ! 1 1 2 2 X12 2 k b1 ¼sin u 2 ksho X1 Bc Zo þ X1 Bc Zo kseo ksho 2 2 8 2Zo se 1 þ cos u X12 Bc kseo ksho 2 X1 2 1 1 2 ksho b2 ¼ sin u 2 kseo X1 Bc Zo þ X12 B2c Zo kseo 2 8 1 2 þ cos u X12 Bc Ksco 4 A.4. System and load data The 500 kV High Dam-Cairo transmission system has the following data [10,16]. Vs ¼ 500 kV MVA ¼ 2100 MW Line reactance, X1 ¼ 0:3020 V/km Line susceptance, Bc ¼ 3:9 £ 1026 s/km
Line length, L ¼ 788 km Generator: 0 0 Xd ¼ 0:275 pu; Xq ¼ 0:275 pu; Xd ¼ 0:05 pu; Tdo ¼ 2:8 s: M ¼ 29:0 Excitation: Ka ¼ 50 and Ta ¼ 0:02 s: Initial operation point: P ¼ 1:0 pu Q ¼ 0:4 pu Vt ¼ 1:0 pu
References [1] St Clair HP. Practical concepts in capability and performance of transmission lines. AIEE transmission (power apparatus and system) paper 53-338 presented at the AIEE specific general meeting, Vancouver, BC, Canada; September 1–4 1953. [2] Dunlop RD, Gutman R, Marchenko PP. Analytical development of loadability characteristics for EHV and UHV transmissions lines. IEEE, Trans PAS-98 1979;606–17. [3] Kay TW, Sauer PW, Smith RA. EHV and UHV line loadability dependence on VAR supply. IEEE Trans Power Apparatus Syst 1982; PAS-101(9):3568–75. [4] Indulkar CS, Viswanathan B, Venkata SS. Reactive Power Constrained Loadability Limits Of Series Capacitor Compensation EHV transmission lines. IEEE Trans Power Syst 1987;2(3):585 –91. [5] Indulkar CS, Viswanathan B, Venkata SS. Maximum power transfer limited by voltage stability in series and shunt compensated schemes for AC transmission systems. IEEE Trans Power Deliv 1989;4(2):1246–52. [6] Miller TJE. Reactive power control in electric systems. New York: Wiley; 1982. [7] Ashok Kumar BS, Parthasarathy K, Prabhakara FS, Khincha HB. Effectiveness of series capacitor in long distance transmission lines. IEEE Trans PAS-89 1970;941–51. [8] Rajaramain R, Alvarado F, Maniaci A, Camfield R, Jalali S. Determination location and amount of series compensation to increase power transfer capability. IEEE Trans Power Syst 1998;13(2):294–300. [9] Woodford DA, Tranawecky MZ. Compensation of long distance AC transmission lines by shunt connected reactance controllers. IEEE, Trans PAS-94 1975;655–64. [10] EL-Marsafawy M. Application of series capacitor and shunt reactor compensation to an existing practical AC transmission line. IEE ProcC 1991;138(4):330–6. [11] Iliceto F, Cinieri E. Comparative analysis of series and shunt compensation schemes for AC transmission system. IEEE Trans PAS-96 1977;1819–30. [12] Steveson WD. Element of power system analysis. New York: McGraw-Hill; 1982. [13] Yu YN. Electric power system dynamic. India: Academic Press Inc; 1983. [14] EL-Emary A, EL-Arini M, EL-Alaily A. Fast contingency evaluation method for generator outage simulation and power system security assessment. ETEP 1995;5(2):129–32. [15] Stagg GW, El-Abiad AH. Computer methods in power system analysis. New York: McGraw-Hill; 1968. [16] FRCU 830202. University Linkage Project, Cairo University. Dynamic equivalent models and decentralized multi level control of unified power system of Egypt. Report 2; 1984. [17] Okamura M, O-ura Y, Hayashi S, Uemura K, Ishiguro F. A new power flow model and solution method including load and generator characteristics and effects of system control devices. IEEE Trans Power Apparatus Syst 1975;PAS-94(3):1042–50. [18] IEEE Task Force on Load Representation for Dynamic Performance, Load representation for dynamic performance analysis. IEEE Trans Power Syst 1993;8(2):472– 82.