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C4 -103
CIGRE 2012
An active angle stability circuit reference for power line resistance-induced
voltage singularities
F.M. EL-SADIK
University of Khartoum
Sudan
SUMMARY
Results of an algebraic solution to the steady state stability limit (SSSL) that has led to
observations of chaotic behaviours in radial systems are presented. The analysis is based on a
different interpretation of roots in a connecting relationship that will describe continuous
power-maximization load constituents; where these will necessarily include the gamut of
maximum powers from zero up to a maximum of the maximums while the associated power
factors are traversed from lagging to leading accordingly. This uses an auxiliary circuit
configuration at the receiving end of a line that will simulate an active angle stability circuit
reference; but will have its variables not showing in the final transfer characteristics of the
line. From among the many observations afforded, some are reported for the influence of line
resistance. These include the depiction of a double-valued function for the implicit reactive
source variables that will in turn evolve into maximum power boundaries, where the lower
may relate to the phenomenon of reduced-generation voltage avalanche with lagging powerfactor loads and the upper to the stressed line scenarios of voltage instability and voltage
collapse with both types of loads. The explicit reactive source circuit variables may also
analyse saturated conditions in a single-machine infinite-bus system or the transfer behaviour
in voltage-constrained lines. The second and most crucial effect of line resistance has evolved
in a newly-introduced solution algorithm for the necessary and sufficient conditions of anglestability while checking the status of sending and receiving-end voltage constraints. This is
manifested in the inducement of possible singularity states within the continuous analytical
function that may account for many of the unexplained radial feeder events. These are
characterized by the occurrence of strong regenerative powers in-reverse that are initiated by a
sudden phase-reversal in auxiliary reactive source reference; and, equally well, by the
tendency of quick recovery to normal function continuity for values in the immediate
neighbourhood of the instigating implicit source variables. The singularity cases reported are
the high resistance type that will not satisfy sending voltage constraint, and the low resistance
type that will satisfy the constraint but at reverse powers of the same ordered magnitude as
those intended for forward transmission. From among the known configurations analysed as
possible singularities are those of the perfect coupling node and the flat transmission voltage
profile. In general, as the algorithm would reveal a larger proportion discontinuities according
fmsadik@yahoo.com
to step size of instigation variables used and the relative amount of resistance involved,
rendering pattern prediction impossible, it is possible to make the stipulation that the various
forms of voltage instability and voltage collapse are manifestations of the inability to sustain
states of maximum received powers, however small these powers may be; as this sustenance
could be violated by resistance-induced discontinuities. However, the continuous analytical
pattern of stability bounds afforded may provide for exact initial guesses in computer
solutions algorithms for voltage stability and the stability of loads. Algebraic equations as
well as graphical depictions for the resistance effects are presented for further observations.
KEYWORDS
Active angle-stability reference, Power-maximizing function, SSSL Singularities.
1. INTRODUCTION
The analysis of the steady-state stability limit as determined by the maximum power-point of
the basic radial system configuration has been an active area of interest; as economic and
environmental constraints for transmission expansion, instigating increased advocating of
compensation schemes, are dictating power systems operation closer to their nose point of the
P-V curve; which requires precise computation of the steady state stability limit [1-4].
This paper introduces the characteristics of a power-maximizing load function at the V-bus of
systems described by E = V + IZ, where E is constant. The analytical constituents of the Vdependent generalized load have evolved in an algebraic formulation to the SSSL problem in
voltage-constrained radial systems in the presence of resistance [5-9]. While the resulting
equations for sending and receiving voltages have on an individual basis provided conclusions
about the reactance behaviour, the approach presented in this paper constitutes a different
interpretation of polynomials roots that has allowed introduction of an angle-stability-based
reference frame whereby the influence of resistance is further emphasized. This consists of
referring all voltage and current variables to voltage VR of an active voltage-sourced circuit
(Figure 1)with the composition of reactive elements shown, where N is introduced as a twomachine circuit simulator variable in an algorithm for the necessary and sufficient SSSL
conditions while checking the status of sending and receiving-end voltage constraints.
Fig. 1: Power Line Angle-Stability Simulator Reference
[E=1.0pu, V1=V; Z1=R+jX1, Z2=jNX3, Z3= j (1-N) X3]
1
In the present context, while the limits (0-1) for N and X3=0, VR=0 will necessarily describe
singularity conditions of perfect coupling nodes, a configuration of the limiting state as N
tends to zero yielded a relationship between V, X3 and VR that will eventually evolve in
equivalent V-bus power-maximizing loads. The explicit relationship connecting the three
stated variables may also analyse conditions for the adjusted reactance of synchronous
machines or the transfer behaviour of compensated transmission lines.
Whereas the proposed algorithm presents a different statement of equations based on Miss
Clark's diagram for the purely reactive network [1], it is the inclusion of resistance in an
algebraic formulation that has afforded two important observations of stability behaviour. The
first is concerned with the introduction of power function continuous boundaries, where the
lower may account for the reduced-generation voltage avalanche phenomenon with lagging
power factor loads and the upper for the known forms of stressed systems instabilities with
both types of loads. The second is related to the inducement of discontinuities in the otherwise
continuous function. These are generally characterized by regenerative reverse powers, of the
same-ordered magnitude as the intended forward powers, which are initiated by phasereversal of induced reference voltage VR; and, at the same time, by the tendency of quick
recovery to normal boundary conditions for values in the immediate neighbourhood of the
instigating variables. As a result, it is possible to identify two types of singularities according
to the relative amount of resistance involved. These are the low resistance type for which
SSSL voltage constraints are approximately upheld and the high-resistance type that will not
satisfy these constraints.
As it is stipulated that the power-maximizing function is a generalized statement for voltage
instability, the stability of loads and of synchronous machines, the considered cases for
graphical and analytical depictions use parameters as those stated in the literature for the
separate stability investigations. However, because of prohibitively large space requirements,
partly symbolic presentations are given for cases that will (a) verify graphical depictions of
function continuities (b) provide for algebraic expressions of stability bounds as exact initial
guesses for numerical solution algorithms and (c) demonstrate the existence of transmission
singularities that may account for many of the unexplained radial feeder events.
2. DEPICTIONS FOR THE POWER-MAXIMIZING ANALYTICAL FUNCTION
The voltage-constrained stability functions are initially obtained in a relationship connecting
sectional impedance parameters of the 3-node configuration for N=1.0. This can be expressed
as a polynomial equation in any one of circuit variables so that, for a given V-bus voltage V,
the stage is set for a dynamic relationship between reference VR and associated X3 such that,
within limits of the relationship, the angle of V with respect to VR, the current angle with
respect to V and subsequently the active and reactive V-bus load constituents will come out as
a result.
The graphical depictions given for this relationship for the cases shown (Figures 2 and 3) will
verify the critical transmission angle (δc) constraint for maximum bus-VR(hence bus-V)
received powers, viz., 𝛿𝑐  tan 1 ( X T / R ) , where, XT = X1+X3.
2
7
6
Reference Reactance X3 (p.u)
Lagging PF Loads
5
Unity PF states
4
3
2
1
0
0
0.5
1
1.5
Active Reference Voltage VR (p.u)
Leading PF Loads
2
2.5
Fig. 2: Angle Stability Reference Functions of evolving V-bus Loads
[
V=0.8501, Z1= 1.0750 p.u.
.................
- - - -V =0.8501, Z1=0.0000+j0.9661pu
V=0.7636, Z1= 1.0750 ]
The curves in Figure (2) are plotted for the evolving roots of V in a VR polynomial expression
that would exactly match those obtained through charts preparations for VR=1.2pu, as
described by the cited criterion for the stability of loads in Reference [1], viz., 0.7636pu and
0.8501pu. However, it can be seen that four values of X3 in the new reference frame analysis
would correspond to the stated VR=1.2pu, and that only one of these would satisfy the
constraint X3=0.8pu at V= 0.7636pu.
It must be stated that whereas the equation roots that result in the power factor modes for the
cases shown are generated in a single function for the purely reactive line, those for the
resistive line would require for their depiction two equations, with the result of double-valued
stability information for X3. Table (1) presents a sample of power-maximizing conditions of
leading and lagging loads stabilities for the case of specified V=0.8501pu.
3
TABLE (1) SAMPLE V-BUS STABILITY INFORMATION FOR V=0.8501pu, Z1= 1.075 p.u
0
X3
P
Q
E
VR
X3
P
Q
E
VR
5.5081
0
0.1312
-1
0
5.6
-0.3309
-0.0237
-0.8113
2.4007
5.6
-0.0098
0.1289
-0.9949
0.0645
5.5999
0.3309
-0.0237
-1
-2.4008
5.5999
0.0098
0.1289
-1
-0.0644
5.6001
0.3309
-0.0237
-1
-2.4007
5.6001
0.0098
0.1289
-1
-0.0646
5
0.37
-0.0556
-1
-2.4742
5.8
0.0316
0.1234
-1
-0.2161
4
-0.4272
-0.1101
-0.7474
2.4314
6
0.0548
0.1169
-1
-0.3872
3.9999
0.4272
-0.1101
-1
-2.4314
6.2
-0.0801
0.109
-0.9577
0.5867
4.0001
0.4272
-0.1101
-1
-2.4314
6.1999
0.0801
0.109
-1
-0.5866
3
0.4825
-0.1739
-1
-2.2455
6.2001
0.0801
0.109
-1
-0.5868
2
0.5411
-0.2589
-1
-1.9364
6.4
0.11
0.0985
-1
-0.8352
1
0.6066
-0.3927
-1
-1.4935
6.6
0.1554
0.0801
-1
-1.2276
0.8
-0.6205
-0.4314
-0.5991
1.3852
6.6
0.2167
0.0501
-1
-1.7446
0.7999
0.6205
-0.4314
-1
-1.3851
6.4
0.2565
0.0273
-1
-2.0355
0.8001
0.6205
-0.4314
-1
-1.3852
6.2
-0.2804
0.0121
-0.8428
2.1824
0.4
0.6472
-0.533
-1
-1.1422
6.1999
0.2804
0.0121
-1
-2.1825
0.1
0.6619
-0.6469
-1
-0.9295
6.2001
0.2804
0.0121
-1
-2.1824
0.01
0.6633
-0.6926
-0.9999
-0.8583
6
0.2995
-0.0008
-1
-2.2805
0
-0.8124
-0.6981
-0.7362
0.8501
5.8
0.316
-0.0126
-1
-2.3503
0.001
0.6627
-0.6976
-0.9991
-0.8509
It must also be noted that some points in the prescribed calculation steps reveal discontinuities
that recover to normal function behaviour, as demonstrated by the neighbouring increments of
instigating X3 shown. These singularity states have evolved in a test algorithm for function
existence while checking the status of specified voltage constraints, as to be presented.
3. ANALYTICAL RESULTS FOR STABILITY BOUNDS
As the complete form of source variables connecting relationship would require prohibitively
large space, partly symbolic depictions generated in MATLAB tool-box with reference to
curves (Figure 3) for R and V specifications are presented for the typical X1 = 1.0pu.
4
30
Reference Reactance X3 (p.u)
25
20
15
10
R
5
0
V
0
1
2
3
4
5
6
7
Reference Voltage V R (p.u)
8
9
10
Fig. 3: Active Stability Reference Relations for X1 =1.0 p.u
[- - - V=0.9pu, R=0.150, 0.250, 0.350, 0.400pu]
[
R=0.2pu, V=0.875, 0.900, 0.925, 0.950pu]
The following equation with coefficients in Appendix (1) is derived in a polynomial radicand
expression for the existence of VR, and hence for any power transfers blow the corresponding
maximums, as dictated by the upper X3limitsand the VR at which they occur:
𝐴𝑋34 + 𝐵𝑋33 + 𝐶𝑋32 + 𝐷𝑋3 + 𝐸 = 0
(1)
The complete equation form would show that the corresponding VR could only be zero for the
condition R=0, as is verified by the curves and by the Clark's equations for reactance
networks. It must also be noted that whereas the describing function may involve large values
for induced VR and associated X3, these are eventually implied into equivalent V-bus powermaximizing constituents.
On the other hand, the bounds to maximum powers in terms of upper VR limits would require
for their assessment the solution of a higher-order polynomial equation, whereby a 3-node
configuration with a flat-voltage profile is demonstrated as a transmission singularity state of
zero power transfers.
In addition, an equation with coefficients as in Appendix (2) that represents the starting point
of zero-maximums lagging loads corresponding to VR=0 in the Figures is obtainable as
follows:
𝐴𝑋36 + 𝐵𝑋35 + 𝐶𝑋34 + 𝐷𝑋33 + 𝐸𝑋32 + 𝐹𝑋3 + 𝐺 = 0
(2)
5
4. AN ALGORITHM FOR THE ANALYTICAL DISCONTIUITIES
The general plan of obtaining necessary and sufficient conditionsfor thestability of a twomachine systemwith internal voltage constraints (Figure 1), upon which the active-reference
simulator model is based, would require a general formulation for the sending and receivingend voltages in the following general terms,where the angular separation between E and VR is
known 'a priori':
E  E (VR ,V1 ,V2 , Z 1 , Z 2 , Z 3 )
(3)
VR  VR ( E , V1 , V2 , Z 1 , Z 2 , Z 3 )
The validity of specified V1 and V2 in the evolving E and VR equations can now be checked
against values obtained using the computed E and VR based on the known critical angular
separation between the two variables. The tolerance used in this presentation for the stringent
test between specified and computed is the infinitesimally small (0.0000001pu).
Expressed in terms of described circuit sectional impedance elements, the set of equations
would collect into two simple equations with N-dependent coefficients as shown in Appendix
(3) for the presentable case X1=X3=V= 1.0pu. The following verification points may be stated
for the described E-constraint test algorithm:
a) For all values of N and R=0, the two equations will reduce to a form that is obtainable
using the Clark's diagram for the stability analysis of purely-reactive systems [1],
b) For N=1, the set of equations reduces to one equation in VR for a specified E, upon
which the original formulation has been based.
c) Either of the two equations can stand on its own in describing continuous transfer
characteristics under stability limits, as has been previously reported for the reactance
behavior in the presence of resistance of radial lines and synchronous machines [6-8].
d) The discontinuities in load function that has demonstrated a distinct phase-reversal in
VR can be shown a consequence of different roots choice in the algorithm for
constraint test.
Table (2) presents results of the connecting solution algorithm for the stated case and R=0.1pu
using N as variable, where V2 is derived on the basis of V1=1.0pu as stated in (c) above.
TABLE (2) SAMPLE V-BUS SINGULARITIES FOR X1=X3=V=1.0pu, R=0.1pu
N
P
Q
E
VR
.9
.99
.999
.9999
.99999
.999999
1
-0.6530
-0.7967
0.8073
0.8083
0.8084
0.8084
-0.8084
-0.8508
-0.6077
-0.5844
-0.5821
-0.5819
-0.5819
-0.5819
-0.6061
-0.8044
-0.9983
-0.9998
-1.0000
-1.0000
-0.8226
1.8694
1.7854
-1.7774
-1.7766
-1.7765
-1.7765
1.7765
In general, whereas the continuity pattern of transfer behavior is generally upheld, the
6
stringent constraint test would reveal discontinuities of a larger proportion according to
calculation steps of instigation variables used and the relative amount of resistance involved.
For example, it can be demonstrated that for V=1.0pu, an impedance of 0.5+j1.0pu would
show no sign of stability over the X3 range (1.00-7.10) for a 0.1pu step size, and over the
range (1.00-2.22) for a 0.01pu step size. However, partial stability is exhibited for the
remaining of the two ranges. On the other hand, an increment of 0.0001 around the 0.5pu
resistance would show partial stability over the two ranges.
It must also be stated that the solution algorithm for SSSL conditions may also involve
resistive elements for the active angle stability simulator circuit that would evolve in a
different configuration of singularity states.
5. CONCLUSIONS
The need for the precise computation of stability limits as determined by the maximum power
transfer capabilities has intensified and the paper presented results of a generalized statement
for the constituents of power-maximizing continuous function using variables of a connected
angle-stability reference simulator circuit. The algebraic formulation has allowed depiction of
stability information that could not have been observed through equations without the
resistance element. In addition, an algorithm aiming at necessary and sufficient conditions for
functions existence has revealed resistance-induced chaotic voltage behaviors as manifested
in discontinuities that may account for some of the many unexplained radial line events. The
sudden phase-reversal in prescribed voltage magnitudes and the associated strong powers in
reverse characterizing the singularity states could be attributed to a different roots choice in
the test algorithm for voltage constraints. In general, whereas the continuity pattern of transfer
behavior is generally upheld, the stringent constraint test would reveal discontinuities of a
larger proportion according to the step size of instigation variables and the relative amount of
resistance involved. However, the well-defined pattern afforded of easily-generated analytical
bounds can serve the purposes of exact initial guesses in computer solution algorithm for the
voltage stability of loads.
BIBLIOGRAPHY
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Kimbark, E.W., Power System Stability; Vol. III, John Wiley and Sons, Second Printing, 1962.
Weedy, B.M., Voltage Stability of Radial Power Links, Proc. IEE, Vol. 115, No. 4, April 1968.
Kundur, P., Power System Stability and Control, Mc Graw Hill, New York, 1994.
Xiao-Ping Zhang, Christian Rehtanz, Bikash Pal, Flexible AC Transmission Systems: Modelling
and Control, Springer, 2006.
EL-Sadik, F.M., Results of Algebraic Equation for the Steady-State Stability Limits in VoltageConstrained Transmission Systems, Proc. CIGRE SC C4 2009 Kushiro Colloquium, June 7-13,
2009, Kushiro, Japan.
EL-Sadik, F.M., An Equation for the Coupling Resistance Influence on (The Many) Radial Link
Power Reserves, Proc. INREC 10, 1ST International Nuclear & Renewable Energy Conference,
March 21-24, 2010, Amman, Jordan.
EL-Sadik F.M., Resistance Influence on the Synchronous and Compensating Reactances in a
SMIB Power System; Proc. SPEEDAM 2010, Power Electronics Electrical Drives Automation
and Motion, 2010 International Symposium on, Pisa, Italy.
EL-Sadik, F.M.: A Relationship for the reactance behavior in voltage-constrained power systems
components, Proc. APPEEC 2011, Asia-Pacific Power and Energy Engineering Conference,
2011, Wuhan University, China.
7
APPENDICES
Appendix (1) Coefficients Equation (1) for Upper-bound X3 in VR Radicand for X1=1.0pu
𝐶 =
𝐴 = (𝑉 2 𝑅 2 − 1 + 𝑉 2 )
𝐵 = (−2 + 4 𝑉 2 𝑅 2 − 2 𝑅 2 + 4 𝑉 2 )
−2 𝑅 2 + 8 𝑉 2 𝑅 2 − 1 − 𝑅 4 + 2 𝑉 2 𝑅 4 + 6 𝑉 2
𝐷 = (4 𝑉 2 𝑅 4 + 4 𝑉 2 + 8 𝑉 2 𝑅 2 )
𝐸 = 𝑉 2 𝑅6 + 𝑉 2 + 3 𝑉 2 𝑅2 + 3 𝑉 2 𝑅4
Appendix (2) Coefficients Equation (2) for the Condition VR = 0, for X1=1.0pu
𝐴 = 1 − 𝑉2 2
𝐵 = (−4 𝑉 2 1 − 𝑉 2 + 2 1 − 𝑉 2 2 )
𝐶 = −2 𝑅 2 + 1 𝑉 2 1 − 𝑉 2 + 4 𝑉 4 − 8 𝑉 2 1 − 𝑉 2 + 1 − 𝑉 2 2 𝑅 2 + 1
𝐷 = (4 (𝑅 2 + 1) 𝑉 4 − 8 (𝑅 2 + 1) 𝑉 2 (1 − 𝑉 2 ) + 8 𝑉 4 )
𝐸 = ( 𝑅2 + 1 2 𝑉 4 + 8 (𝑅 2 + 1) 𝑉 4 + (−2 (𝑅 2 + 1) 𝑉 2 (1 − 𝑉 2 ) + 4 𝑉 4 ) (𝑅2 + 1))
𝐹 = 6 𝑅2 + 1 2 𝑉 4
𝐺 = 𝑅2 + 1 3 𝑉 4
Appendix (3) Simultaneous Equations Coefficients for Connecting Solution Algorithm
AE 4 + BE 2 + C = 0
𝐴 = ((−2 (𝑅2 + 1) 𝑅 1 − 𝑁 2 − (𝑅 2 + 4 𝑁 + 1 − 𝑁 2 ) (−2 (4 − 𝑅 2 ) 𝑅 − 2 (𝑅 2 + 1) 𝑅
+ 8 𝑅)) ((−2 (4 − 𝑅 2 ) 𝑅 − 2 (𝑅 2 + 1) 𝑅 + 8 𝑅) 1 − 𝑁 2 + 2 𝑅 1 − 𝑁 2 )
2
− 𝑅 2 + 1 1 − 𝑁 2 − 𝑅 2 − 4 𝑁 − 1 − 𝑁 2 (𝑅 2 + 4))
𝐵 = ((−2 (𝑅2 + 1) 𝑅 1 − 𝑁 2 − (𝑅 2 + 4 𝑁 + 1 − 𝑁 2 ) (−2 (4 − 𝑅 2 ) 𝑅 − 2 (𝑅 2 + 1) 𝑅
+ 8 𝑅)) (−(−2 (4 − 𝑅 2 ) 𝑅 − 2 (𝑅 2 + 1) 𝑅 + 8 𝑅) (𝑅 2 + 4) 𝑉2 2
+ 2 𝑅 1 − 𝑁 2 (−𝑅 2 − 4)) − 2 (−(𝑅2 + 1) (𝑅 2 + 4) 𝑉22 − (𝑅 2 + 4 𝑁
+ 1 − 𝑁 2 ) (−𝑅2 − 4)) ((𝑅 2 + 1) 1 − 𝑁 2 − 𝑅 2 − 4 𝑁 − 1 − 𝑁 2 ) (𝑅 2
+ 4))
2
𝐶 = − − 𝑅 2 + 1 𝑅 2 + 4 𝑉22 − 𝑅 2 + 4 𝑁 + 1 − 𝑁 2 −𝑅 2 − 4
(𝑅 2 + 4)
𝑆𝑉𝑅4 + 𝑈𝑉𝑅2 + 𝑊 = 0
𝑆 = ((−2 𝑅 1 − 𝑁 2 − (−2 (4 − 𝑅 2 ) 𝑅 − 2 (𝑅 2 + 1) 𝑅 + 8 𝑅) 1 − 𝑁 2 ) ((𝑅 2 + 4 𝑁
+ 1 − 𝑁 2 ) (−2 (4 − 𝑅 2 ) 𝑅 − 2 (𝑅 2 + 1) 𝑅 + 8 𝑅) + 2 (𝑅 2
+ 1) 𝑅 1 − 𝑁 2 ) − 𝑅 2 + 4 𝑁 + 1 − 𝑁 2 − 𝑅 2 + 1 1 − 𝑁 2 2 (𝑅2 + 4))
𝑈 = ((−2 𝑅 1 − 𝑁 2 − (−2 (4 − 𝑅 2 ) 𝑅 − 2 (𝑅 2 + 1) 𝑅 + 8 𝑅) 1 − 𝑁 2 ) (−(−2 (4
− 𝑅 2 ) 𝑅 − 2 (𝑅 2 + 1) 𝑅 + 8 𝑅) (𝑅 2 + 4) 𝑉22 + 2 𝑅 1 − 𝑁 2 (−𝑅 2 − 4))
− 2 (−(𝑅2 + 4) 𝑉22 − 1 − 𝑁 2 (−𝑅 2 − 4)) (𝑅 2 + 4 𝑁 + 1 − 𝑁 2 − (𝑅 2
+ 1) 1 − 𝑁 2 ) (𝑅 2 + 4))
2
𝑊 = − − 𝑅 2 + 4 𝑉22 − 1 − 𝑁 2 −𝑅 2 − 4
(𝑅 2 + 4)
8