Voltage Constraints for the Maximum Power Transfer Theorem

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Voltage Constraints for the Maximum Power
Transfer Theorem & Brainerd’s Resonance Voltage Impedance
Condition
*
Fayez Mohammed EL-Sadik
ABSTRACT
It is shown that Brainerd’s resonance voltage impedance relation and the well-known
maximum power transfer impedance conditions of conjugate matching are two states
of unstable equilibrium with the terminal voltages of a matched linear transducer
circuit directly related to that of Brainerd’s. The relationship is based on stability
indices derived from internal impedance and voltage constraints with the result that
additional singularity states that may apply to radial power and communication links
and for which no reference can be found in the literature are observed. These include
the conditions for maximum source voltage; voltage profiles with inter-nodal
constraints and the source and load voltage magnitudes evolving in states of zero load
powers and a maximum of the maximums of transmitted powers.
Keywords: maximum power, maximum voltage, radial power links, equilibrium states
1.
INTRODUCTION
The theorems for the maximum load voltage
and maximum power transfer of systems that
can be represented by the circuit of Figure 1
were formulated on the basis of fixed
magnitudes of either source voltage E or
terminal voltage V according to the application
for which the circuit is intended. In electronic
engineering where E is normally fixed, the two
theorems evolve, as a result, in the impedance
variations of a passive load circuit with the
occurrence of maximum power of E2/4R, when
ZL=Z*sand, according to Brainerd [1],
maximum voltage when:
Figure 1: Series Representation of a Voltage
ZL = -jZs2/Xs. Notice that the two conditions
share the occurrence of a common δ between
E and the resulting V that is identical to the
transmission angle γ = tan-1 Xs/Rs, with no
reference made as to any constraints on the
magnitude of E. On the other hand, in power
systems engineering where V, e.g., can be
specified such as to be found in radial power
links, the following well-known expression for
the receiving end power derived with Vreference gives δc = γ = tan-1 Xs/Rs, as the first
maximum with no upper limit to this
maximum set by any other magnitude of E
thereafter:
VE
V2
P=
sin(δ + α ) −
sin α
Zs
Zs
,
α = tan
Link / Terminated Transducer Circuit
*
−1
(1)
Rs
Xs .
Because of the high powers involved, the lack
of a clear analytic solution to the capability
limits of power systems components has
Department of Electrical Engineering, University of Khartoum, Sudan; fmsadik@yahoo.com
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57
Figure 2: Transmission Circuit with Inter-Voltage and Sectional Impedances Constrains
prompted criteria for practical stability limits,
e.g., thermal limits of transmission lines and
the rotors of synchronous machines.
In this paper, evidence of the existence of a
current equilibrium state imposing constraints
on the source and load voltages of a linear
transducer circuit under maximum power
transfer conditions is presented. The nonlinear relationship connecting the two
variables is derived from an analytic solution
to the problem of stability limits of a radial
power link with resistance and inter-voltage
constraints, Figure 2); giving two indices one
of which will define an equilibrium state of a
maximum no-load voltage Vmax that
corresponds to Brainerd’s load impedance
induction for this maximum. As a result it is
shown that the conditions ZL=-jZ2s/Xs for
maximum voltage and ZL=Z*s, for maximum
load power have unique sets of unstable
operational voltages that are directly related to
that of Brainerd’s. In addition, and as a
consequence of the resulting voltage stability
boundary curves, other equilibrium states that
may relate to radial power and communication
links and for which no reference can be found
in linear circuit theory are observed. These
include the conditions for maximum source
voltage, voltage profiles under inter-nodal
constraints and the source and load voltage
magnitudes evolving in states of zero load
powers and a maximum of the maximums of
transmitted powers.
2.
RESULTS OF THE POWERMAXIMIZING RELATION
The load impedances for maximum voltage
and maximum power of a terminated
transducer circuit are obtainable as two special
cases of the following generalized current
58
equilibrium relation where β, Vmax are stability
indices derived from internal impedance and
voltage constraints;
I eq =
β
Z
2
Vmax
−V 2 + j
2
s
Xs
V
Z s2
(2)
The relationship is power maximizing in that
the resulting source voltage E = V + IZ
occurs at δ = δc for all β, Vmax & V.
The corresponding receiving-end maximum
of powers are obtained from the in-phase
component as:
Pmax =
βV
Z
2
s
2
Vmax
−V 2
with the corresponding
component given by:
QPmax =
(3)
leading
reactive
Xs 2
V
Z s2
(4)
The statement for equilibrium current Ieq is
based on the results of an analytic solution to
the stability problem of a radial power link
with resistance and inter-voltage constraints
for which, as concluded by Kimbark some
four decades ago [2], no algebraic equations
have been derived. While solution details and
practical implications require the space of a
full publication, validity of the stated indices is
verified herein for the present purposes of
introducing the general shape of the resulting
V-Pmax and E-Pmax stability boundary curves
that will embody Brainerd’s load impedance
induction for Vmax together with the necessary
and sufficient conditions of conjugate
matching. This uses for a case study a line
with unity resistance and the sectional
impedances defined by the internal voltage
magnitude constraints shown in Figure 3;
Sudan Engineering Society JOURNAL, August 2005, Volume 51 No.44
giving a total impedance Zs=1.0 + jXs .
The general solution algorithm for this twonode system evolves in a constraint
relationship connecting the two variables Va
and Vb at the maximum power transfer
condition of δ = δc. A state of no-load
terminal voltage is then identified as a limiting
condition for the existence of Va for any
specified magnitude of Vb. This equilibrium
state is taken as the voltage stability index Vmax
which, together with β in Ieq given by (2), are
determined by Vb and the sectional
impedances constraints. For the present study
case, taking Vb = 1.0 p.u, the two indices relate
to those in Figure 3 as follows:
Vmax =
Z s2
β
(5)
β = Z − Xs
2
s
For any given Xs, therefore, Vmax and β are
identified to yield the simultaneous conditions
δ = δc and Vb = 1.0 p.u. for all V ≤ Vmax . This
gives the desired stability boundary curves a
sample of which is shown plotted in Figure 4
for the case Zs=1.0+j3.0 p.u. where A, B, C,
D, F, G & H represent the operational voltages
and powers of seven impedance equilibrium
states.
While two of the states (A, B) will identify
with the impedance conditions for maximum
voltage and maximum power of a terminated
transducer circuit, the others relate to newly-
introduced impedance and voltage states under
which neither a terminated transducer circuit
nor a radial voltage link will be able to sustain
stable equilibrium as can be seen from the
following.
3.
EQUILIBRIUM
VOLTAGES
AND IMPEDANCE STATES
The conditions δ = tan-1Xs and Vb = 1.0
p.u. are satisfied for all Xs with the
substitution in Ieq of the terminal voltage
magnitudes V shown in table I using β,
Vmax for the case Zs=1.0 + jXs considered
in the configuration of Figure 3). The
corresponding source voltage magnitudes
E together with the equilibrium impedance
ZL for each state will come out as shown in
Table 1.
The following identifications can be made
for the seven states of unique equilibrium
sets of source and load voltages which are
in direct proportionality to Vmax and with
their terminal impedances line parameterdictates while circuit internal voltage
constraint for the configuration used is
being observed.
State (A): Equilibrium No-Load Voltage State:
This can be seen as evolving in Brainerd’s
impedance induction for load voltage
resonance except that the source is constrained
to the magnitude show0n.
Figure 3: Case Study Configuration; ZS = 1.0 + j XS
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59
Figure 4: Case Study Equilibrium Voltage Stability Boundaries Zs=1.0+j3.0 p.u.
State (B): Voltages of Conjugate-Matching:
While the impedance derived for this state
concurs with that of a matched linear
transducer circuit, no reference can be found
for the associated source voltage magnitude as
representing a unique state of unstable
equilibrium.
State (C): A Condition of Maximum of the
Maximums of Received Powers:
This condition can be deduced from the
maximum power expression (3) to give in
conjunction with the corresponding E an upper
limit to the maximums of received powers.
The condition is synonymous with conjugate
matching for values of Xs that will satisfy β =
1.0
State (D) : A Maximum Source Voltage
Condition:
This newly introduced condition for linear
transducer circuits is the dual of Brainerd’s
maximum load voltage condition with the
terminal voltage V duly constrained to the
magnitude shown. As for the case of upper
maximums of powers stated in (C), the
60
condition is also synonymous with conjugate
matching for values of Xs that will satisfy β =
1.0
States (F) & (G): Conditions for Zero Load
Voltages and Powers:
In addition to the no-load state due to Vmax and
the associated E derived in (A), a source
voltage indicated by state (F) with the
magnitude shown evolves in two conditions
one of which gives rise to a zero-load voltage
and hence zero load power. The other state (G)
that is associated with the terminal impedance
and voltage relations shown can be seen as
reducing to the equilibrium no-load state (A)
for values of β = 1.0
State (H) : Voltage Profile Predictions With
Inter-Nodal Constraints:
In power systems engineering where the
voltage profile under steady-state stability
limits with inter-voltage constraints is of
primary concern, equation (3) readily solves
for the condition in terms of V for any
specified maximum power with the
corresponding E coming out as a result.
Alternatively, the voltages and power limits at
Sudan Engineering Society JOURNAL, August 2005, Volume 51 No.44
Table 1: Equilibrium Voltages and Impedance States
State
V
A
Vmax
β
B
β 2 +1
D
β 2 +1
− jZ s2
β 2 +1
Vmax
Zs
β
0
G
2β
Vmax
β 2 +1
Zs
β
Zs
β
( Z s − 1) + β
2
2
Vmax
a given profile can be obtained; e.g., a flat
voltage profile indicated by the crossover state
(H) in Figure 4 is governed by the impedance
and voltage relations shown.
4.
CONCLUSION
Evidence of the existence of a constraint
relationship connecting E & V under
maximum power transfer conditions of
systems described by E = V + IZ has been
presented. The analysis uses internal voltages
and sectional impedance constraints resulting
in a well-known power system configuration
problem for which an analytic solution in
terms of steady state stability limits has long
been sought. The sample of results given
describes P-V & P-E stability curves for radial
links that will embody known as well as newly
introduced impedance conditions for linear
terminated transducer circuits. The complete
Xs
Z *s = 1 − jX s
Vmax
⎛ β +1 ⎞
⎜
⎟
⎜ 2 Z ⎟Vmax
s ⎠
⎝
F
H
1
Vmax
Zs
2β
Zs β 2 +1
2
1
ZL
Vmax
Vmax
C
E
Z s2
( β − jX s )
β 2 + X s2
Z s2
( β 2 − jX s )
4
2
β + Xs
Vmax
Vmax
--
Vmax
2Z s2
β 2 − 1 + j2 X s
β
( Z s − 1) + β
2
2
Vmax
Zs
Xs
−j
2
2( Z s − 1)
solution algorithm for the power limits of
radial systems with resistance and multi nodal
constraints will afford better insight into
design optimization problems as to be seen in
a future full publication.
REFERENCES
1. Brainerd, J.G., “Some General Resonance
Relations and A Discussion of Thevenin’s
Theorem”; Proceedings of the Institute of
Radio Engineers, Volume 21, Number 7,
July 1933.
2. Kimbark, E.W., “Power System Stability;
Vol. III”, John Wiley and Sons, Second
Printing , 1962.
3. Fayez Mohamed El-Sadik, “Towards the
prediction of Power System Voltage
Collapse”; Sudan Engineering Society
Journal, Volume 47, No.38, July 2000.
Sudan Engineering Society JOURNAL, January 2006, Volume 52 No.45
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