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. 13
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g½Pg ; V; ; B° ފ < 0 is not shown in Equation (7.64) because it is added to L°x; kÞ only
when variables go outside limits.
The SVC susceptance, B( ), may be expressed as either an equivalent susceptance, Bsvc,
or a susceptance that is an explicit function of the SVC ¬ring angle, . Both parameters may
be used as state variables and, respectively, form the basis of the two SVC models addressed
in this section.
The contribution of the SVC to the Lagrangian function is explicitly represented in
Newton™s method as an equality constraint given by the following equation:

Lsvc °x; kÞ ¼ qk Qk : °7:65Þ

In this expression, x is the vector of state variables, ½Vk B° ފt ; Qk is the reactive power
injected or absorbed by the SVC at bus k, as given in Equation (5.5); and k is the vector of
Lagrange multipliers, with qk being the Lagrange multiplier at bus k associated with the
reactive power balance equation. The variable B( ) is either Bsvc or , depending on the
SVC model used in the OPF study.



7.7.2 Linearised System of Equations

Representation of the SVC controller into the OPF algorithm using Newton™s method
requires that for each SVC present in the network, matrix W be augmented by one row and
one column. Either or Bsvc, depending on the SVC models selected, enters as an extra state
variable in the OPF formulation.
Application of Newton™s method to the SVC ¬ring-angle model is given by the following
linearised equation:

2 32 3 2 3
ÁVk rVk
q2 L qQk q2 L
6 qV 2 qVk q 76 7 6 7
qVk
6 76 7 6 7
k
6 76 7 6 7
6 qQk qQk 76 7 6 7
76 Áq k 7 ¼ À6 rq k 7: °7:66Þ
6 0
6 qVk q 76 7 6 7
6 76 7 6 7
4 q2 L 54 5 4 5
qQk qL
2

Á r
q qVk q q 2

The entries in Equation (7.66) are obtained by deriving Equation (7.65) with respect to the
relevant state variables and Lagrange multipliers. These terms are given in explicit form in
Appendix B, Section B.3. The derivative terms corresponding to inequality constraints are
not required at the beginning of the iterative solution; they are introduced into matrix
Equation (7.66) after limits become enforced in response to limits violations.
An alternative OPF model for the SVC is readily established by choosing the SVC
equivalent susceptance, Bsvc, to be the state variable rather than the ¬ring angle, . The
293
STATIC VAR COMPENSATOR

linearised system of equations describing the alternative SVC OPF model is:
2 32 3 2 3
ÁVk rVk
q2 L qQk q2 L
6 qV 2 qVk qVk qBsvc 76 7 6 7
6 76 7 6 7
k
6 76 7 6 7
6 qQk qQk 76 7 6 7
76 Áq k 7 ¼ À6 rq k 7: °7:67Þ
6 0
6 qVk qBsvc 76 7 6 7
6 76 7 6 7
4 q2 L 54 5 4 5
qQk
0 ÁBsvc rBsvc
qBsvc qVk qBsvc
The entries in Equation (7.67) are obtained by deriving Equation (7.65) with respect to the
relevant state variables and Lagrange multipliers. These terms are given in explicit form in
Appendix B, Section B.3.
In OPF studies it is normal to assume that voltage magnitudes at SVC terminals
are controlled within limits (e.g. 0.95“1.05 p.u). However, more stringent voltage magnitude
requirements are met with ease in Newton™s method. For instance, to control the voltage
magnitude at bus k at a ¬xed value, it is necessary only to add the second derivative term of
a large, quadratic, penalty factor to the second derivative term of the Lagrangian function
with respect to the voltage magnitude Vk (i.e. q2 L=qVk ). Also, the ¬rst derivative term of the
2

quadratic penalty function is added to the corresponding gradient element (i.e. qL=qVk ).
Hence, in Equations (7.66) and (7.67) the diagonal elements corresponding to voltage
magnitude Vk will have a very large (in¬nite) value, resulting in a null voltage increment
ÁVk.
The SVC is well initialised by selecting a ¬ring-angle value corresponding to the
equivalent reactance resonance peak, which can be calculated using Equation (5.39). The
SVC Lagrange multiplier, qk, is initialised at zero value. These initial values give rise to
´
very robust iterative solutions (Ambriz-Perez, 1998).

7.7.3 Static VAR Compensator Test Cases

The ¬ve-bus system in Section 7.3.6 is modi¬ed to include one SVC at Main, as shown in
Figure 7.6. The objective is to minimise its active power generation cost. The SVC
capacitive and inductive reactance are XC ¼ 0.9375 p.u., and XL ¼ 0.1625 p.u., respectively,
The lower and upper limits for the ¬ring angle are 90 and 180 , respectively. The initial
¬ring angle is given a value ¼ 145 .
Four case studies are carried out: cases A and B use the SVC model based on the ¬ring-
angle concept, whereas cases C and D use the model based on the equivalent variable
susceptance. Moreover, cases A and C consider the voltage magnitude at Main to be allowed
to take any value in the range 0.95“1.1 p.u; cases B and D consider the voltage magnitude at
Main to be ¬xed at 1.1 p.u.

7.7.3.1 Firing-angle model

In the ¬ring-angle model, two cases are simulated:
 Case A: the voltage magnitude at Main is allowed to take any value in the range 0.95“
1.1 p.u.;
 Case B: the voltage magnitude at Main is ¬xed at 1.1 p.u.
294 OPTIMAL POWER FLOW

40 + j5
45 + j15
2.31
80.14


Lake Main
North 14.91
32.98 32.27 14.93


47.15 1.15 1.16 7.77
2.72 10.04 12.06

30.17
5.08 0.09
27.65
1.61
2.23
46.79 5.08
30.63
28.05
1.36
5.06 5.66
2.76


Elm
South 55.98 54.93


20 + j10 60 + j10
4.33
3.92
87.87 4.71

Modi¬ed ¬ve-bus network with one static VAR compensator, and optimal power ¬‚ow
Figure 7.6
solution


The power ¬‚ow results are shown in Figure 7.6. The voltage magnitudes and phase
angles and Lagrange multipliers are given in Table 7.9. Similar results are given for case B
in Table 7.10. The SVC susceptance values and reactive power injections are shown in
Table 7.11.
As expected, active power generation cost and active power loss increase in case B. The
OPF results are 748.339 $ hÀ1 , 3.14226 MW, and 37.13 MVAR. In this case there are
relatively large ¬‚ows of reactive power from bus Main to other buses, thus increasing
network losses.


Table 7.9 Nodal voltages in the modi¬ed network: case A (use of static VAR
compensator model based on the ¬ring-angle concept, with voltage magnitude
at Main allowed to take any value in the range 0.95“1.1 p.u.)
Bus
””””””””””””””
”””””””””””””””
Elm Main Lake South North
Voltage:
Magnitude (p.u.) 1.075 1.085 1.083 1.100 1.109
À 4.450 À 3.962 À 3.701 À 1.304
Phase angle (deg) 0.000
lp °$ MWÀ1 hÀ1 Þ 4.2625 4.2324 4.2217 4.1030 4.0411
295
STATIC VAR COMPENSATOR

Table 7.10 Nodal voltages in the modi¬ed network: case B (use of static VAR
compensator model based on the ¬ring-angle concept, with voltage magnitude
at Main ¬xed at 1.1 p.u.)
Bus
””””””””””””””
”””””””””””””””
Elm Main Lake South North
Voltage:
Magnitude (p.u.) 1.080 1.100 1.095 1.100 1.111
À 4.471 À 4.148 À 3.836 À 1.2613
Phase angle (deg) 0.000
lp °$ MWÀ1 hÀ1 Þ 4.2650 4.2431 4.2299 4.1024 4.0426




Static VAR compensator parameters at each iteration: ¬ring-angle model, cases A and B
Table 7.11
Case A Case B
(deg) (deg)
Iteration Beq (p.u.) Q (MVAR) Beq (p.u.) Q (MVAR)
À 51.420 À 51.420
0 145.000 0.514 145.000 0.514
À 7.231 À 6.630
1 136.627 0.056 136.598 0.054
À 14.534 À 60.640
2 137.819 0.131 144.712 0.501
À 11.213 À 37.130
3 137.234 0.095 140.832 0.306
À 12.061
4 137.347 0.102 “ “ “
“ Iteration not required; model has converged.




The results in Table 7.11 indicate that in order to maintain the voltage magnitude at Main
at 1.1 p.u. it is necessary for the SVC to inject more reactive power. It should be noted that
the minus sign indicates injection of reactive power. These results illustrate the strong
convergence of the SVC OPF algorithm, with solutions achieved in 4 and 3 iterations,
respectively.


7.7.3.2 Susceptance model

The SVC modelled in the form of a susceptance replaces the SVC ¬ring-angle-based model
used in the two test cases above (cases A and B). The initial SVC susceptance value is set at
Bsvc ¼ 0.514 p.u., which corresponds to ¼ 145 . Two cases are simulated:
 Case C: the voltage magnitude at Main is allowed to take any value in the range 0.95“
1.1 p.u.;
 Case D: the voltage magnitude at Main is ¬xed at 1.1 p.u.
Convergence is obtained in four and three iterations for cases C and D, respectively. As
expected, the solution for voltage magnitude, voltage phase angle, active and reactive power
generation, and Lagrange multipliers coincide with those presented in Tables 7.9 and 7.10.
296 OPTIMAL POWER FLOW

Table 7.12 Equivalent static VAR compensator susceptances for cases C and D
Case C Case D
Iteration Bsvc (p.u.) Q (MVAR) Bsvc (p.u.) Q (MVAR)
À 51.420 À 51.420
0 0.514 0.514
À 7.204 À 6.594
1 0.056 0.054
À 14.534 À 60.641
2 0.131 0.501
À 11.213 À 37.130
3 0.095 0.306
À 12.061
4 0.102 “ “
“ Iteration not required; model has converged.




The equivalent susceptance values taken by the SVC model during the iterative process
are shown in Table 7.12. It can be observed from Tables 7.11 and 7.12 that both sets of SVC
susceptances coincide with each other.

7.8 THYRISTOR-CONTROLLED SERIES COMPENSATOR

This section studies the topic of OPF TCSC modelling and simulation (Acha and Ambriz-
´
Perez, 1999). This is done within the context of Newton™s method in which the TCSC is
modelled as an adjustable, nonlinear series reactance which is a function of the TCSC ¬ring
angle.

7.8.1 Lagrangian Function

The constrained optimisation problem, stated in generic form in Equation (7.1), is converted
into an unconstrained problem. This involves formulating a suitable Lagrangian function for
the TCSC controller, which may take the following form:
L°x; kÞ ¼ f °Pg Þ þ kt h½Pg ; V; ; X° ފ: °7:68Þ
In this expression, f °Pg Þ is the objective function; h½Pg ; V; ; X° ފ represents the power
¬‚ow equations; x is the vector of state variables; k is the vector of Lagrange multipliers for
equality constraints; and Pg , V, , and X( ) are the active power generation, voltage
magnitude, voltage phase angle, and TCSC reactance, respectively. The reactance, X( ),
is an explicit function of the TCSC ¬ring angle, . The inequality constraint
g½Pg ; V; ; X° ފ < 0 is not shown in Equation (7.68) because it is added to L°x; kÞ only
when variables go outside limits.
The power ¬‚ow mismatch equations at buses k and m are explicitly modelled in the
Lagrangian function as an equality constraint given by the following equation:
Ltcsc °x; kÞ ¼ pk °Pk þ Pdk À Pgk Þ þ qk °Qk þ Qdk À Qgk Þ
þ pm °Pm þ Pdm À Pgm Þ þ qm °Qm þ Qdm À Qgm Þ: °7:69Þ
In this expression, Pdk, Pdm, and Qdk, Qdm are the active and reactive power loads at buses k
and m; Pgk, Pgm, Qgk, and Qgm are the scheduled active and reactive power generations at
buses k and m; and pk, pm, qk, and qm are Lagrange multipliers at buses k and m.
297
THYRISTOR-CONTROLLED SERIES COMPENSATOR

XC m l
k
Pml Rml + jXml
XL


Figure 7.7 Compensated transmission line


As shown in Figure 7.7, the active power ¬‚ow across branch m“l, Pml, is controlled by the
TCSC connected between buses k and m. In the OPF formulation this operating condition is
expressed as an equality constraint, which remains active throughout the iterative process
unless one expressly wishes this constraint to be deactivated.
The Lagrangian function, L, of the total branch kÀl, may be expressed by:
L ¼ Ltcsc °x; kÞ þ Lflow °x; kÞ; °7:70Þ
where
Lflow ¼ ml °Pml À Pspecified Þ: °7:71Þ
In this expression, ml is the Lagrange multiplier for the active power ¬‚ow in branch mÀl,
and Pspeci¬ed is the target active power ¬‚ow through the TCSC controller.


7.8.2 Linearised System of Equations

Incorporation of the TCSC controller into the OPF algorithm using Newton™s method
requires that for each TCSC present in the network, matrix W be augmented by two rows
and two columns when the aim is to exert active power ¬‚ow control. However, if the TCSC
is not controlling active power ¬‚ow then matrix W is augmented only by one row and one
column. The former case uses the Lagrange multiplier, ml, to account for the contribution
of the power ¬‚ow through branch m“l, and enters as an extra state variable in the OPF
formulation.
Application of Newton™s method to the TCSC ¬ring-angle model is given by the
following linearised equation:
2 32 3 23
Ázk
Wkk Wkm 0 Wk gk
6W 76 7 6g 7
6 mk Wmm Wml Wm 76 Ázm 7 6 m7
7 ¼ À6 7: °7:72Þ
6 76
40 Wml Wll Wl 54 Ázl 5 4 gl 5
Áz
W k W m W l W g
In this expression, the structure of matrix and vector terms Wkk, Wkm, Wmk, Wmm, Ázk,
Ázm, gk, and gm is given by Equations (7.25)“(7.32), respectively. The additional matrix
terms in Equation (7.72) re¬‚ect the contribution of , the TCSC state variable. These terms
are given explicitly by:
22 3
qL q2 L qPk qQk
W k ¼ Wtk ¼ 4 qk q qVk q q q 5; °7:73Þ
0 0 0 0
298 OPTIMAL POWER FLOW

2 3
qL qL qPl qQl
2 2
6 7
6 qm ql qm qVl qm qm 7
62 7
6 qL qQl 7
q2 L qPl
6 7
6 qVm 7;
¼ 6 qVm ql qVm qVl qVm 7 °7:74Þ
Wlm ¼ Wtml 6 qP 7
qPm
6 7
m
6 07
0
6 ql 7
qVl
6 7
4 qQm 5
qQm
0 0
ql qVl
2 3
q2 L q2 L qPm qQm
6 qm q qVm q 7
q
6 7
W m ¼ Wtm ¼6 7; °7:75Þ
4 qP 5
qPml
ml
0 0
qm qVm
2 3
q2 L q2 L qPl qQl
6 q2 ql qVl ql ql 77
6
6 7
l
6 q2 L qPl qQl 7
qL2
6 7
6 7
6 qV q qVl qVl 7
qVl2
7; °7:76Þ
Wll ¼ 6 l l
6 qP 7
qPl
6 7
l
6 07
0
6 ql 7
qVl
6 7
4 qQl 5
qQl
0 0
ql qVl
2 3
0 0 00
W l ¼ Wtl ¼ 4 qPml qPml 5; °7:77Þ
00
ql qVl
22 3
qL
W ¼ 4 q 2 0 5; °7:78Þ
0 0

Áql Št ;
Ázl ¼ ½ Ál ÁVl Ápl °7:79Þ

rql Št ; °7:80Þ
gl ¼ ½ rl rVl rpl

Áml Št ;
Áz ¼ ½ Á °7:81Þ
rml Št :
g ¼ ½ r °7:82Þ

The ¬rst and second partial derivatives for the various entries in Equation (7.72) are derived
from the Lagrangian function of Equation (7.70) and are given in Appendix B, Section B.4.
The derivative terms corresponding to inequality constraints are entered into matrix W
only if limits are enforced as a result of one or more state variable having violated
299
THYRISTOR-CONTROLLED SERIES COMPENSATOR

limits. It should be noted that the procedure in Equation (7.72) corresponds to the case
when the TCSC is controlling active power ¬‚owing through branch m“l (standard control
mode).
In OPF applications, minimum-cost solutions are obtained when the OPF algorithm itself
selects the optimum level of power ¬‚ow through the TCSC. However, any change in the
TCSC operating mode is easily accommodated in the OPF formulation given in Equation
(7.72). For instance, if the TCSC is not controlling active power ¬‚ow then matrix W
and vector g are suitably modi¬ed to re¬‚ect this operating mode. This can be achieved by
adding the second derivative term of a large (in¬nite) quadratic penalty factor to the
diagonal element of the matrix in Equation (7.78) corresponding to multiplier ml,
thus forcing this multiplier to be zero for the whole of the iterative process. The ¬rst
derivative term of the quadratic penalty function is added to the corresponding element in
Equation (7.80).
The Lagrange multipliers for active and reactive power ¬‚ow mismatch equations are
initialised at the p value given by the lossless economic dispatch solutions and at q equal
to 0, respectively. For TCSC Lagrange multipliers the initial value of ml is set to zero.
Experience shows that these values give rise to very robust iterative solutions (Ambriz-
´
Perez, 1998). The main factor affecting the OPF rate of convergence of TCSC-upgraded
networks is the initial ¬ring angle, . Good starting conditions are required to prevent the
solution diverging or arriving at some anomalous value. Good initial conditions for the
TCSC ¬ring angle were established in Section 5.8.3. Use of Equations (5.72)“(5.73)
invariably leads to good OPF solutions for TCSC-upgraded networks.


7.8.3 Thyristor-controlled Series Compensator Test Cases

The ¬ve-bus test system of Section 7.3.6 is used to study the impact of the TCSC on the
network. The TCSC is added in series with transmission line Lake“Main, and the dummy
bus LakeTCSC is added to enable such a connection to take place.
The OPF solution is achieved in ¬ve iterations to a mismatch tolerance of 1e À 9 and
starting from a TCSC ¬ring-angle value equal to 150 . The TCSC optimises the active
power ¬‚ow level in transmission line Lake“Main to a value of 14.97 MW. Moreover, the
OPF solution yields the following minimum active power generation cost and network
losses: 747.975 $ hÀ1 and 3.05 MW, respectively. The TCSC capacitive and inductive
reactance values required to achieve the result are: XC ¼ 0.9375 % and XL ¼ 0.1625 %,
respectively, using a base voltage of 400 kV. The optimal power ¬‚ows are shown in
Figure 7.8. The nodal voltage magnitudes and phase angles and the Lagrange multipliers are
given in Table 7.13.
It can be observed that the OPF solution changes little compared with the base OPF case
presented in Section 7.3.6 when no TCSC is used. This may be explained by the fact that the
solution achieved in Section 7.3.6 is already a very good solution and that the OPF is ¬xing
the level of compensation afforded by the TCSC to be fairly small.
The TCSC ¬ring angles, per iteration, are shown in Table 7.14, highlighting the very
strong convergence characteristics of OPF using Newton™s method and the importance of
selecting good initial conditions. For completeness, the equivalent TCSC reactance is also
provided.
300 OPTIMAL POWER FLOW

40 + j5
45 + j15
0.29
80.15


Lake
North Main
14.95
32.98 32.26 14.97

Lake-TCSC
1.03
47.17 1.33 2.12
5.16 4.37
5.03 2.19
27.70 30.08

5.44 4.92
5.19
46.80
30.55
28.11
1.89
5.01 3.52
1.59


Elm
South 56.04 54.98


20 + j10 60 + j10
6.47
6.10
87.89 14.40

Modi¬ed ¬ve-bus system, and optimal power ¬‚ow solution
Figure 7.8


Table 7.13 Nodal voltages in TCSC-upgraded network

Bus
”””””
””””””””””” ””””””””””
” ””””””””””””””
LakeTCSC Elm Main Lake South North
Voltage:
Magnitude (p.u.) 1.078 1.072 1.077 1.078 1.100 1.109
À 3.534 À 4.417 À 3.846 À 3.622 À 1.303
Phase angle (deg) 0.000
lp °$ MWÀ1 hÀ1 Þ 4.2232 4.2639 4.2341 4.2232 4.1031 4.0412



Thyristor-controlled series compensator (TCSC)
Table 7.14
parameters

TCSC parameters
(deg)
Iteration XTCSC (p.u.)
À 0.0180
0 150.000
À 0.0169
1 150.587
À 0.0101
2 162.845
À 0.0130
3 154.328
À 0.0119
4 156.399
À 0.0119
5 156.407
301
UNIFIED POWER FLOW CONTROLLER

7.9 UNIFIED POWER FLOW CONTROLLER

The UPFC OPF model presented in this section enables very ¬‚exible and reliable power
´
system optimisation studies to be carried out (Ambriz-Perez et al., 1998). The ¬‚exibility
stems from the generality of the UPFC model and the robustness from the strong
convergence exhibited by the OPF solution using Newton™s method. The UPFC model may
be set to control active and reactive powers simultaneously as well as nodal voltage
magnitude, at either the sending or the receiving end bus. Alternatively, the UPFC model
may be set to control one or more of the parameters above in any combination or to control
none of them.


7.9.1 Uni¬ed Power Flow Controller Lagrangian Function

Based on the equivalent circuit shown in Figure 5.17 and Equations (5.50)“(5.59), the
Lagrangian function for the UPFC may be written as:
L°x; kÞ ¼ f °Pg Þ þ kt h°Pg ; V; ; cR ; VcR ; vR ; VvR Þ: °7:83Þ
In this expression, f °Pg Þ is the objective function to be optimised; h°Pg ; V; ;
VcR ; cR ; VvR ; vR Þ represents the power ¬‚ow equations; x is the vector of state variables;
k is the vector of Lagrange multipliers for equality constraints; and Pg , V, and  are the
active power generation, voltage magnitude, and voltage phase angle, respectively. The
UPFC control variables are cR, VcR, vR , and VvR . The inequality constraints g°Pg ; V; ;
VcR ; cR ; VvR ; vR Þ < 0 are not shown in Equation (7.83) because it is added only to L°x; kÞ
when there are variables outside limits.
The Lagrangian function, Lkm(x,k), corresponding to the power ¬‚ow mismatch equations
at buses k and m, is given by the following equation:
Lkm °x; kÞ ¼ pk °Pk þ Pdk À Pgk Þ þ qk °Qk þ Qdk À Qgk Þ
þ pm °Pm þ Pdm À Pgm Þ þ qm °Qm þ Qdm À Qgm Þ: °7:84Þ
In this expression Pdk, Pdm, Qdk, and Qdm are the active and reactive power loads at buses k
and m; Pgk, Pgm, Qgk, and Qgm are the scheduled active and reactive power generations at
buses k and m; and pk, pm, qk, and qm are Lagrange multipliers at buses k and m. The
vector of state variables x is [V d]t, where V and d include both nodal voltages and UPFC
voltage sources.


7.9.2 Direct-current Link Lagrangian Function

A fundamental premise in the UPFC model is that the active power supplied to the shunt
converter, PvR , must satisfy the active power demanded by the series converter, PcR. This
condition must be met throughout the solution process. In the OPF formulation this condition
is expressed as an equality constraint,
Lsh--se °x; kÞ ¼ sh--se °PvR þ PcR Þ; °7:85Þ
where sh“se is the Lagrange multiplier associated with the shunt and series power
converters.
302 OPTIMAL POWER FLOW

m l
k
Rml + jXml
Pml


Qml



Uni¬ed power ¬‚ow controller power ¬‚ow constraint at bus m
Figure 7.9


7.9.3 Uni¬ed Power Flow Controller Power Flow Constraints

The power injected at bus m by the UPFC, as illustrated in Figure 7.9, can be formulated as
a power ¬‚ow constraint in the branch connecting buses m and l. We may write:
Lml °x; kÞ ¼ p ml °Pml À Pspecified Þ þ q ml °Qml À Qspecified Þ; °7:86Þ
where p ml and q ml are, respectively, the Lagrange multipliers associated with the active
and reactive power injections at bus m; and Pspeci¬ed and Qspeci¬ed are, respectively, the
speci¬ed active and reactive powers leaving bus m.
In conventional OPF formulations, such constraints are enforced only if power ¬‚ow limits
have been exceeded. However, in this particular application this constraint may remain
active throughout the iterative solution.
The UPFC Lagrangian function comprising the individual contributions presented above
is as follows:
Lupfc °x; kÞ ¼ Lkm °x; kÞ þ Lsh--se °x; kÞ þ Lml °x; kÞ: °7:87Þ


7.9.4 Linearised System of Equations

Incorporation of the UPFC controller into the OPF algorithm using Newton™s method
requires that, for each UPFC, matrix W be augmented by up to eleven rows and columns.
This procedure corresponds to the case where the UPFC is operated in standard control
mode (i.e. it is controlling the nodal voltage magnitude at bus k, active power ¬‚owing from
buses m to l, and reactive power injected at bus m). The linearised system of equations for
minimising the UPFC Lagrangian function of Equation (7.87), using Newton™s method is:
2 32 32 3
Ázk Àgk
Wk k Wk m Wk cR Wk vR Wk shÀse 0 0
6 76 76 7
Wm vR Wm shÀse Wm l Wm mÀl 76 Ázm 7 6 Àgm 7
6 Wm k Wm m Wm cR
6 76 76 7
6 WcR k 0 76 ÁzcR 7 6 ÀgcR 7
WcR m WcR cR WcR vR WcR shÀse 0
6 76 76 7
6 76 76 7
0 76 ÁzvR 7 ¼ 6 ÀgvR 7:
6 WvR k WvR m WvR cR WvR vR WvR shÀse 0
6 76 76 7
6 WshÀse k WshÀse m WshÀse cR WshÀse vR 0 76 ÁzshÀse 7 6 ÀgshÀse 7
0 0
6 76 76 7
6 76 76 7
Wl l Wl mÀl 54 Ázl 5 4 Àgl 5
40 Wl m 0 0 0
ÁzmÀl ÀgmÀl
0 WmÀl m 0 0 0 W mÀll 0
°7:88Þ
303
UNIFIED POWER FLOW CONTROLLER

In this expression, the structure of matrix and vector terms Wkk, Wkm, Wmk, Wmm, Dzk,
Dzm, gk, and gm is given by Equations (7.25)“(7.32), respectively. Also, Wml, Wlm, Wll, Dzl,
and gl are given by Equations (7.74), (7.76), (7.79), and (7.80), respectively. The additional
matrix terms in Equation (7.88) re¬‚ect the contribution of cR, VcR, vR , and VvR , the UPFC
state variables. These terms are given explicitly by:
2 3
q2 L q2 L qPk qQk
6 7
6 qk qcR qVk qcR qcR qcR 7
WcR k ¼ Wk cR ¼ 6 7; °7:89Þ
t
4 q2 L qPk qQk 5
q2 L
qk qVcR qVk qVcR qVcR qVcR
2 3
q2 L q2 L qPk qQk
6 7
6 qk qvR qVk qvR qvR qvR 7
WvR k ¼ Wk vR ¼ 6 7; °7:90Þ
t
4 q2 L qPk qQk 5
q2 L
qk qVvR qVk qVvR qVvR qVvR
!
qPshÀse qPshÀse
WshÀse k ¼ Wtk shÀse ¼ 0; °7:91Þ
0
qk qVk
2 3
q2 L q2 L qPm qQm
6 7
6 q q qVm qcR qcR qcR 7
WcR m ¼ Wm cR ¼ 6 m2 cR 7; °7:92Þ
t
4 qL qQm 5
q2 L qPm
qm qVcR qVm qVcR qVcR qVcR
2 3
q2 L q2 L qPm qQm
6 7
6 q q qVm qvR qvR qvR 7
WvR m ¼ Wm vR ¼ 6 m2 vR 7; °7:93Þ
t
4 qL qQm 5
q2 L qPm
qm qVvR qVm qVvR qVvR qVvR
!
qPshÀse qPshÀse
WshÀse m ¼ Wtm shÀse ¼ 0; °7:94Þ
0
qm qVm
2 3
qPml qPml
0 07
6 q qVm
¼6 m 7;
WmÀlm ¼ °7:95Þ
WtmmÀ l 4 qQml 5
qQml
0 0
qm qVm
2 3
q2 L q2 L
6 qcR qVcR 7
qcR
2
6 7
¼6 7; °7:96Þ
WcR cR
4 qL 5
q2 L 2

qVcR qcR qVcR
2

2 3
q2 L q2 L
6 7
6 q q qcR qVvR 7
¼ 6 cR2 vR
WcR vR ¼ WtvR cR 7; °7:97Þ
4 qL q2 L 5
qVcR qvR qVcR qVvR
304 OPTIMAL POWER FLOW

!
qPshÀse qPshÀse
WshÀse cR ¼ WtcR shÀse ¼ 0; °7:98Þ
0
qcR qVcR
2 3
q2 L q2 L
6 7
qvR qVvR 7
qvR
6 2
¼6 7; °7:99Þ
WvR vR 6 7
4 qL 5
q2 L 2

qVvR qvR qVvR2
!
qPshÀse qPshÀse
WshÀse vR ¼ ¼ ; °7:100Þ
WtvR shÀse
qvR qVvR
2 3
qPml qPml
0 07
6 q qVl
6 7
l
Wlm-- l ¼ WtmÀ ll ¼6 7; °7:101Þ
4 qQml qQml 5
0 0
ql qVl


ÁVcR Št ;
ÁzcR ¼ ½ ÁcR °7:102Þ
ÁVvR Št ;
ÁzvR ¼ ½ ÁvR °7:103Þ
ÁzshÀse ¼ ½ÁshÀse Š; °7:104Þ
Áq ml Št ;
ÁzmÀl ¼ ½ Áp ml °7:105Þ
rVcR Št ;
gcR ¼ ½ rcR °7:106Þ
rVvR Št ;
gvR ¼ ½ rvR °7:107Þ
gshÀse ¼ ½rshÀse Š; °7:108Þ
rq ml Št :
gmÀl ¼ ½ rp ml °7:109Þ


The elements of matrix W are given explicitly in Appendix B, Section B.5. The deri-
vative terms corresponding to inequality constraints are not required at the beginning of
the iterative process; they are introduced into matrix Equation (7.88) only after limits
become enforced.
The representation given in Equation (7.88) corresponds to a situation where the UPFC is
operated in standard control mode. However, if different UPFC operating modes are
required then matrix W and multipliers k are modi¬ed with ease to re¬‚ect the new operating
mode.
For instance, if buses m and k are PQ type and the UPFC is not controlling active power
¬‚owing from buses m to l and reactive power is not injected at bus m then matrix W and
gradient vector g are modi¬ed as follows: (1) the second derivative term of a large (in¬nite),
quadratic penalty factor is added to the diagonal elements of matrix Wll, corresponding to
the multipliers p ml and qml; (2) the ¬rst derivative terms of the quadratic penalty functions
are evaluated and added to the corresponding gradient elements in gl. Alternatively, if only
one operating constraint is released, say reactive power injected at bus m, then only the
diagonal element of matrix Wll corresponding to multiplier qml is penalised.
305
UNIFIED POWER FLOW CONTROLLER

The Lagrange multipliers for active and reactive power ¬‚ow mismatch equations are
initialised at the p value given by the lossless economic dispatch solution and q equal to
zero, respectively. For UPFC Lagrange multipliers, the initial value of sh“se is set to p, and
p m“l and q m“l are set equal to zero. Experience has shown that these values give rise to
very robust iterative solutions. Equations for initialising the voltage magnitudes and phase
angles of the series and shunt sources are given in Section 5.8.4.


7.9.5 Uni¬ed Power Flow Controller Test Cases

One UPFC is added to the ¬ve-bus system of Figure 7.3, in series with the transmission line
Lake“Main. A dummy bus, termed LakeUPFC, is added to enable the UPFC model to be
connected, as shown in Figure 7.10. The UPFC is used to maintain active and reactive power
at 25 MW and À 6 MVAR, respectively, at the sending end of transmission line LakeUPFC“
Main. The shunt converter is used to maintain Lake™s nodal voltage magnitude at 1 p.u.
The two UPFC voltage sources are initialised with reference to equations and guidelines
given in Section 5.8.4, resulting in the following values: VcR ¼ 0.025 p.u., cR ¼ 76.5 ,
VvR ¼ 1.0 p.u., and vR ¼ 0 ; The resistances of the coupling transformers are ignored and
their inductive reactances are taken to be XcR ¼ XvR ¼ 0:1 p.u.. The voltage magnitude VcR
varies in the range 0.001“0.6 p.u., and VvR in the range 0.9“1.1 p.u.
This is a case of regulated UPFC operation, and the OPF solution, albeit optimal, is not
expected to be the one that yields minimum cost. This point will be addressed further, by


40 + j5
45 + j15
1.87
80.15

Lake-
North Lake Main
36.95 35.92 24.93
25.00
UPFC

43.20 4.19 2.32 4.16
4.40 6.00
8.37 2.45
34.07 23.43

5.98 6.70
1.16
42.85
23.77
34.78
3.98 8.31 2.40
3.57


South Elm
52.76 51.68


20 + j10 60 + j10
7.75 7.59
88.47 24.15

UPFC-upgraded ¬ve-bus system, and optimal power ¬‚ow solution
Figure 7.10
306 OPTIMAL POWER FLOW

Table 7.15 Nodal voltages in the UPFC-upgraded network
Bus
”””””
””””””””””” ””””””””””
” ””””””””””””””
LakeUPFC Elm Main Lake South North
Voltage:
Magnitude (p.u.) 1.007 0.999 1.006 1.000 1.029 1.036
À 3.128 À 4.722 À 3.580 À 4.685 À 1.402
Phase angle (deg) 0.000
lp °$ MWÀ1 hÀ1 Þ 4.2680 4.2823 4.2246 4.2680 4.1077 4.0412




numerical example, in Section 7.9.6. The cost and active power losses given by the OPF
solution in this test case are: 750.357 $ hÀ1 , and 3.631 MW, respectively. The optimal power
¬‚ow results are shown in Figure 7.10; the voltage magnitudes and phase angles, and the
Lagrange multipliers are given in Table 7.15.
Compared with the base case shown in Figure 7.3, larger active power ¬‚ows in
transmission lines North“Lake and South“Lake take place in order to meet the demand
imposed by the UPFC power constraints. By comparing both OPF solutions, it can be
observed that in the UPFC-upgraded system there are increases in active and reactive power
generation of 0.5 MW and 7.015 MVAR. Furthermore, the generation cost and the network
losses increased by 2.027 $ hÀ1 and 0.5 MW, respectively. The reason for the higher
generation cost and power loss can be explained in terms of a reduced number of control
variables available to the OPF solution; the UPFC is set to regulate active and reactive
power ¬‚ows and voltage magnitude.
It may be argued, with reference to the voltage information shown in Table 7.15, that
the nodal voltage regulation imposed by the UPFC at Lake yields a much improved voltage
pro¬le than that achieved by the base OPF solution, shown in Table 7.2, where nodal voltage
magnitudes edged on the high side. Conversely, the p values tend to be higher in the present
test case than in the base OPF case, where no UPFC is used. This may be explained in terms
of the slightly higher cost incurred by the regulating action of the UPFC controller.
The voltage magnitudes and phase angles of the UPFC series and shunt voltage sources
are shown in Table 7.16, highlighting the strong convergence characteristic of the OPF using
Newton™s method and the all-important point of selecting good initial conditions for the two
UPFC voltage sources.




Parameters of uni¬ed power ¬‚ow controller voltage sources
Table 7.16
Series source Shunt source
cR (deg) vR (deg)
Iteration VcR (p.u.) VvR (p.u.)
À 76.500
0 0.025 1.000 0.000
À 94.102 À 4.718
1 0.052 0.998
À 94.876 À 4.705
2 0.052 0.997
307
SUMMARY

Table 7.17 Uni¬ed power ¬‚ow controller (UPFC) operating modes

Generation cost ($ hÀ1 )
Operating mode Number of iterations Power loss (MW)
Normal UPFC operation 2 750.357 3.631
Fixed voltage (at bus Lake) 2 749.928 3.519
Fixed P and Q 3 748.236 3.119
All constraints deactivated 3 747.828 3.015




7.9.6 Uni¬ed Power Flow Controller Operating Modes

In order to illustrate the behaviour of the various UPFC operating modes, its functional
constraints are freed in sequence. The normal UPFC operating mode (all constraints
activated) is compared with cases where active and reactive power ¬‚ows are freed, and
the voltage magnitude remains ¬xed; the voltage magnitude is freed, and active and reactive
power ¬‚ows are ¬xed; all three constraints are freed. Table 7.17 presents a summary of the
results.
As expected, the case of normal UPFC operation gives the most expensive solution,
whereas the case where all the constraints are deactivated gives the minimum cost solution.
The former case was studied in Section 7.9.5, and the latter case is very much in line with
the results obtained in the base OPF solution, where no UPFC is used. However, it may be
argued that one of the main purposes of installing a UPFC controller in the ¬rst place is, to
have the ability to regulate power ¬‚ows and voltage magnitude at the point of UPFC
deployment.



7.10 SUMMARY

The OPF algorithm studied in this chapter is a direct application of Newton™s method to the
minimisation of a multivariable, nonlinear function. An iteration of the OPF algorithm
consists of the simultaneous solution of all the unknown variables involved in the problem
using Lagrange functions. Second partial derivatives of the Lagrange function with respect
to all the variables and the Lagrange multipliers are determined and the resultant terms are
suitably accommodated in matrix W. This matrix has a block matrix structure where each
block stores 12 nonzero elements per bus.
The OPF solution gives the optimum operational state of a power network where a
speci¬c objective has been met, and the network is subjected to physical and operational
constraints. Active power generation cost is the most popular objective function used today.
An OPF computer program is an effective tool to conduct power system studies. It
provides a realistic and effective way to obtain a minimum production cost of active power
dispatch within the speci¬ed plant and transmission network operating limits. The optimal
redistribution of generated active power results in a signi¬cant reduction in the active power
generation cost and active power transmission losses.
FACTS controller models have been developed for an OPF algorithm using ¬rst
principles. The models have been linearised and included in the frame of reference afforded
308 OPTIMAL POWER FLOW

by Newton™s method. The extended OPF Newton algorithm is a very powerful tool capable
of solving FACTS-upgraded power networks very reliably, using a minimum of iterative
steps. The computational ef¬ciency of the algorithm is further increased by employing
the multiplier method to handle the binding set.
The FACTS controller models have been shown to be very ¬‚exible; they take into account
their various operating modes as well as their interactions with the network and other
controllable plant components. Flexibility has been achieved without adversely affecting the
ef¬ciency of the solution. In general, the solution of networks with and without FACTS
controllers has been achieved in the same number of iterations. The effect of the initial
conditions on convergence has also been studied. Improper selection of initial conditions
may degrade convergence or, more seriously, cause the solution to diverge.


REFERENCES
´
Acha, E., Ambriz-Perez, H., 1999, ˜FACTS Device Modelling in Optimal Power Flows Using Newton™s
Method™, Proceedings of the 13th Power System Computation Conference, Trondhein, Norway,
June“July 1999, pp. 1277“1284.
´
Acha, E., Ambriz-Perez, H., Fuerte-Esquivel, C.R., 2000, ˜Advanced Transformer Control Modelling
in an Optimal Power Flow Using Newton™s Method™, IEEE Trans. Power Systems 15(1)
290“298.
Alsac, O., Bright, J., Prais, M., Stott, B., 1990, ˜Further Developments in LP-based Optimal Power
Flow™, IEEE Trans. Power Systems 5(3) 697“711.
´
Ambriz-Perez, H., 1998, Flexible AC Transmission Systems Modelling in Optimal Power Flows Using
Newton™s Method, PhD Thesis, Department of Electronics and Electrical Engineering, University of
Glasgow, Glasgow.
´
Ambriz-Perez, H., Acha, E., Fuerte-Esquivel, C.R., De la Torre, A., 1998, ˜Incorporation of a UPFC
model in an Optimal Power Flow using Newton™s Method™, IEE Proceedings on Generation,
Transmission and Distribution 145(3) 336“344.
´
Ambriz-Perez, H., Acha, E., Fuerte-Esquivel, C.R., 2000, ˜Advanced SVC Models for Newton“Raphson
Load Flow and Newton Optimal Power Flow Studies™, IEEE Trans. Power Systems 15(1) 129“136.
Bertsekas, D.P., 1982, Constrained Optimization and Lagrange Multiplier Methods, Academic Press,
New York.
Dommel, H.W., Tinney, W.F., 1968, ˜Optimal Power Flow Solutions™, IEEE Trans. Power Apparatus
and Systems PAS-87(10) 1866“1876.
El-Hawary, M.E., Tsang, D.H., 1986, ˜The Hydro-thermal Optimal Power Flow, A Practical Formula-
tion and Solution Technique using Newton™s Approach™, IEEE Trans. Power Systems PWRS-1(3)
157“167.
Happ, H.H., 1977, ˜Optimal Power dispatch “ A Comprehensive Surrey™, IEEE Trans. Power Apparatus
and Systems PAS-96(3) 841“854.
Huneault, M., Galiana, F.D., 1991, ˜A Survey of the Optimal Power Flow Literature™, IEEE Trans.
Power Systems 6(2) 762“770.
´
IEEE/CIGRE (Institute of Electrical and Electronic Engineers/Conseil International des Grands
reseaux Electriques), 1995, ˜FACTS Overview™, Special Issue, 95TP108, IEEE Service Centre
Piscataway, NJ.
Luenberger, D.G., 1984, Introduction to Linear and Nonlinear Programming, 2nd edn, Addison-Wesley,
New York.
Maria, G.A., Findlay, J.A., 1987, ˜A Newton Optimal Power Flow Program for Ontario Hydro EMS™,
IEEE Trans. Power Systems PWRS-2(3) 576“584.
309
REFERENCES

Monticelli, A., Liu, W.H.E., 1992, ˜Adaptive Movement Penalty Method for the Newton Optimal Power
Flow™, IEEE Trans. Power Systems 7(1) 334“342.
Sasson, A.M., 1969, ˜Nonlinear Programming Solutions for Load-¬‚ow, Minimum-loss, and Economic
Dispatching Problems™, IEEE Trans. Power Apparatus and Systems PAS-88(4) 399“409.
Sasson, A.M., Viloria, F., Aboytes, F., 1973, ˜Optimal Load Flow Solution Using the Hessian Matrix™,
IEEE Trans. Power Apparatus and Systems PAS-92(1) 31“41.
Stagg, G.W., El-Abiad, A.H., 1968, Computer Methods in Power Systems Analysis, McGraw-Hill, New
York.
Sun, D.I., Ashley, B., Brewer, B., Hughes, A., Tinney W.F., 1984, ˜Optimal Power Flow By Newton
Approach™, IEEE Trans. Power Apparatus and Systems PAS-103(10) 2864“2880.
Tinney, W.F., Hart, C.E., 1967, ˜Power Flow Solution by Newton™s Method™, IEEE Trans. Power
Apparatus and Systems PAS-96(11) 1449“1460.
Wood, A.J., Wollenberg, B., 1984, Power Generation, Operation and Control, 2nd edn, John Wiley &
Sons, Chichester.
8
Power Flow Tracing

8.1 INTRODUCTION

Deregulation and unbundling of transmission services in the electricity supply industry
worldwide has given rise to a new area of operation known as ˜electrical energy trading™.
Since the late 1980s, the time of privatisation of the UK supply industry, several proposals
for the operation of the power network have been put forward in various parts of the world.
Arguably, the concept of virtual direct access through a voluntary wholesale pool (Secretary
of State for Energy, 1988) was the ¬rst workable market-oriented operating philosophy, but
it is not in operation any more; it has been superseded by the New Energy Trading
Agreement (NETA), which is in operation in England and Wales (Saunders and Boag,
2001). Furthermore, the NETA operating philosophy will soon be extended to encompass
Scotland, becoming the British Electricity Trading and Transmission Arrangements
(BETTA) (OFGEM, 2003). The ˜Pool™ concept served well the needs of the newly
established market but its management attracted criticism for being too complex to operate
and for being open to market distortions.
In academic circles, a major criticism of the ˜pool™ was that it did not address crucial
issues such as the use of system charges and power transmission losses on a sound
engineering basis. It was also argued that this operating philosophy was limiting business
opportunities, such as the provision of ancillary services. The ˜pool™ was born out of the
inability to trace individual generator power contributions in the network. Indeed, at the time
of privatisation, the issue was deemed as too complicated to have a viable solution.
The electricity pool rules state that ˜˜with an integrated system it is not possible to trace
electricity from a particular generator to a particular supplier™™ (EPEW, 1993). Nevertheless,
it was shown in the mid-1990s that the tracing of power ¬‚ows from generators to suppliers
was indeed possible, and algorithms, based on the concepts of dominant power ¬‚ows and
proportional sharing, were put forward to solve such an outstanding issue. Independent,
basic research at the University of Manchester Institute of Science and Technology (UMIST;
Kirschen and Strbac, 1999; Kirschen, Allan, and Strbac, 1997), Durham University (Bialek,
1996, 1997, 1998), and University of Glasgow (Acha, Fuerte-Esquivel, and Chua, 1996;
Acha et al., 1997) led to similar outcomes. Earlier work had addressed the plausibility of
such a solution, but this work was con¬ned to solving dominant power ¬‚ows in radial
systems (Macqueen, 1993) as opposed to general, interconnected networks. More recently,


FACTS: Modelling and Simulation in Power Networks.
´ ´
Enrique Acha, Claudio R. Fuerte-Esquivel, Hugo Ambriz-Perez and Cesar Angeles-Camacho
# 2004 John Wiley & Sons, Ltd ISBN: 0-470-85271-2
312 POWER FLOW TRACING

variations and further applications of the basic algorithms have been published (Acha, 1998;
Reta and Vargas, 2001), including the incorporation of FACTS equipment models (Acha
et al., 2003; Laguna-Velasco et al., 2001). The notion of proportional sharing has been
shown to be mathematically demonstrable (Laguna-Velasco, 2002).
In this chapter, the power ¬‚ow tracing algorithm put forward in an earlier publication
(Acha et al., 2003) is ¬rst detailed. It should be mentioned that power ¬‚ow tracing is only a
mechanism for tracing generation costs and allocating charges for use of line. The algorithm
is in fact an electricity auditing procedure and answers all questions relating to individual
generator contributions to optimal power ¬‚ows, power losses, and costs in each plant
component of the power network. A distinction is made between generation costs, possibly
attributable to fuel burning, and costs incurred for use of ˜wires™. The contribution of
FACTS equipment to reactive power ¬‚ows and losses is discussed.


8.2 BASIC ASSUMPTIONS

As successfully argued by Reta and Vargas (2000), the power tracing algorithms are based
only on electric circuit concepts and hence, at their core, they use the proportional sharing
principle (Bialek, 1996, 1997, 1998). This is explained with reference to the simple radial
transmission system shown in Figure 8.1, consisting of three buses, two generators, two

B1 B3
B2
100 MW 150 MW 140 MW
160 MW 110 MW
G1
TL 2
T L1
140 MW
50 MW 50 MW
100 MW
L1 L3
L2
G2
(a)

B1 B3
B2
75 MW 70 MW
160 MW 110 MW 100 MW
G1
T L2
T L1
70 MW
50 MW 25 MW

L1 L3
L2
(b )
B1 B3
B2
75 MW 70 MW
0 MW 0 MW

TL2
T L1
70 MW
0 MW 25 MW
100 MW
L1 L3
L2
G2
(c)

Figure 8.1 Individual power ¬‚ows in a simple radial system: (a) power solution; (b) contribution of
generator G1, and (c) contribution of generator G2
313
MATHEMATICAL JUSTIFICATION OF THE PROPORTIONAL SHARING PRINCIPLE

i
Bm
B2
k
150 MW
«P P
P = P ¬ im +
jm
·
Pim
100 MW mk ¬
P + Pjm P +Pjm ·
mk

 im im

50 MW 

Pjm
P
P = P ¬ im + ·
ml ¬
+ Pjm P + Pjm ·
ml
P
 
im im
Pjm
100 MW
j l
(a) (b)

Figure 8.2 The proportional sharing principle: intuitive appeal: (a) The situation at bus B2 and
(b) the situation at bus Bm

transmission lines, and three loads. The power ¬‚ow solution is given in Figure 8.1(a), where
it is appreciated that the combined generation of 260 MW by generators, G1 and G2, go to
supply the system load of 240 MW. Each transmission line in this contrived system incurs
power losses of 10 MW. Figure 8.1(b) shows the contribution of generator G1 to the power
¬‚ows at the sending and receiving ends of transmission lines TL1 and TL2 and to loads L1,
L2, and L3. By the same token, Figure 8.1(c) shows the contribution of generator G2 to the
power ¬‚ows at the sending and receiving end of transmission line TL2 and to loads L2 and
L3. Notice that there is no contribution of generator G2 to load L1 and that it causes no
power loss in transmission line TL1. Hence, the 10 MW loss in TL1 is due entirely to G1,
whereas the 10 MW loss in TL2 is shared equally by G1 and G2.
As appreciated from Figure 8.1(b), generator G1 contributes to power ¬‚ows in branches
TL1 and TL2 and to loads L1, L2, and L3; it is also clear that, in this case, the entire system
falls within the dominion of generator G1. Similarly, as appreciated from Figure 8.1(c),
generator G2 contributes only to the power ¬‚ow in branches TL2 and to loads L2 and L3.
Hence, the dominion of generator G2 is more restricted than that of generator G1.
Furthermore, there is an overlap between the dominions of generators G1 and G2.
Of particular interest are the power in¬‚ows and out¬‚ows in bus B2, where the principle of
proportional sharing is self-evident. An anatomy of this bus may be drawn: Figure 8.2(a)
represents the situation prevailing in bus B2 in Figure 8.1(a), and Figure 8.2(b) is a more
generic expansion of the concept involved. À Á À Á
In Figure 8.2(b), the expressions Pmk Pim Pim þ Pjm and Pmk Pjm Pim þ Pjm represent
the contributions ofÀ in¬‚ows Pim and Pjm À out¬‚ow Á
to mk, respectively. Similarly, the
 Á
expressions Pml Pim Pim þ Pjm and Pml Pjm Pim þ Pjm represent the contributions of
in¬‚ows Pim and Pjm to out¬‚ow ml, respectively.


8.3 MATHEMATICAL JUSTIFICATION OF THE PROPORTIONAL
SHARING PRINCIPLE

The following justi¬cation is drawn from Laguna-Velasco (2002). With reference to
Figure 8.2, the voltage at bus m may be expressed as a function of the branch impedance Zmk
and its current ¬‚ow Imk, or of Zml and Iml :
Vm ¼ Zmk Imk ¼ Zml Iml : °8:1Þ
314 POWER FLOW TRACING

Alternatively, it may be expressed as the product of the equivalent impedance, as seen from
bus m, and the total injected current into bus m:
 
Zmk Zml
Vm ¼ IT ; °8:2Þ
Zmk þ Zml

where

IT ¼ Iim þ Ijm : °8:3Þ

Combining Equations (8.1) and (8.2), and solving for Imk and Iml , gives:
 
Zml
Imk ¼ IT ; °8:4Þ
Zmk þ Zml
 
Zmk
Iml ¼ IT : °8:5Þ
Zmk þ Zml
An expression for the power ¬‚ow in branch mk may be derived as a function of the powers
contributed by in¬‚ows im and jm:
Ã
Smk ¼ Vm Imk
  
Ã
Zml à Ã
¼ Vm à Iim þ Ijm
à °8:6Þ
Zmk þ Zml
 
À Á
Ã
Zml
¼ Sim þ Sjm ;
à Ã
Zmk þ Zml
where
Ã
Sim ¼ Vm Iim ; °8:7Þ
Ã
Sjm ¼ Vm Ijm : °8:8Þ
By the same token, the power ¬‚ow in branch ml is:
 
À Á
Ã
Zmk
Sml ¼ Sim þ Sjm : °8:9Þ
à Ã
Zmk þ Zml
Equations (8.6) and (8.9) can be given in terms of only complex powers as opposed to
powers and impedances by making use of the relations

Zmk ¼ Vm SÃ
2
mk

and Zml ¼ Vm Sml :
2
   
Smk Smk
Smk ¼ Sim þ Sjm ; °8:10Þ
Smk þ Sml Smk þ Sml
   
Sml Sml
Sml ¼ Sim þ Sjm : °8:11Þ
Sml þ Smk Sml þ Smk

It should be noted that the following power conservation relation:

Sim þ Sjm ¼ Smk þ Sml
315
DOMINIONS

can be used instead in Equations (8.10) and (8.11):
 
Sim Sjm
Smk ¼ þ Smk ; °8:12Þ
Sim þ Sjm Sim þ Sjm
 
Sim Sjm
Sml ¼ þ Sml : °8:13Þ
Sim þ Sjm Sim þ Sjm
Separation of the real and imaginary components in Equations (8.12) and (8.13) leads to
useful expressions for active and reactive powers:
 
Pim Pjm
Pmk ¼ þ Pmk ; °8:14Þ
Pim þ Pjm Pim þ Pjm
 
Qim Qjm
Qmk ¼ þ Qmk ; °8:15Þ
Qim þ Qjm Qim þ Qjm
 
Pim Pjm
Pml ¼ þ Pml ; °8:16Þ
Pim þ Pjm Pim þ Pjm
 
Qim Qjm
Qml ¼ þ Qml : °8:17Þ
Qim þ Qjm Qim þ Qjm
Note that the expressions for active power are those derived intuitively in Section 8.3,
appearing in Figure 8.2(b). They are generalised in Sections 8.5“8.6 for the case of n in¬‚ows
and loads.


8.4 DOMINIONS

The concept of dominion is at the centre of the power tracing algorithm. In its most basic
form it may be seen as a directed graph consisting of one source, and one or more sinks. The
set of branches linking source and buses are related to transmission components present in
the network, such as lines, transformers, high-voltage direct-current (HVDC) links and
series FACTS equipment. The directions of the branches are dictated by the power ¬‚ow or
the optimal power ¬‚ow (OPF) solution upon which the tracing study is based.
There are several ways of carrying out the actual implementation of the algorithm used
for determining the sources dominions. Kirschen, Allan, and Strbac (1997) give one
possible course of action, where the concepts of ˜commons™ and ˜links™ are used. A
˜common™ is de¬ned as a set of contiguous buses supplied by the same source. Branches
within a common are termed internal branches™, and the set of external branches linking two
commons is termed the ˜link™. The analysis is conducted at the common and link level ¬rst.
Once the power contribution to each common is known then all buses, loads, and branches
within the common are allocated a share of the power ¬‚owing into that common.
An alternative algorithm is detailed in this section. It is a lower-level algorithm in which
the concepts of source dominions and common branches are used (Bialek, 1997), as
opposed to commons and links (Kirschen, Allan, and Strbac, 1997).
The source dominions are determined as follows:
 Select the ¬rst source and, starting from the source bus, check all the branches with a
connection to the bus.
316 POWER FLOW TRACING

 Branches in which the power ¬‚ows away from the bus (i.e. out¬‚ows) are included as part
of the dominion along with the bus at the receiving end of the branch. Conversely,
branches in which the power ¬‚ows into the bus (i.e. in¬‚ows) do not form part of the
dominion of the source. The procedure is repeated for each new bus as soon as it becomes
part of the dominion of the source.
 After no further buses can be reached, the process comes to a halt, resulting in a directed
subgraph containing only branches that carry power pertaining to the source currently
under analysis.
 The above procedure is repeated for the second source of the network, the third, and so on.
 If the dominion of a source contains no branches, then the dominion is a degenerated
dominion, and the source will contribute power only to the local load.
The use of the branch“node incidence matrix offers a systematic way for implementing
this algorithm. This matrix is highly sparse and yields very ef¬cient solutions.
By way of example, Figure 8.3 shows the ¬ve-bus system with active and reactive power
¬‚ows, which correspond to the optimal power ¬‚ow solution as opposed to a conventional
power ¬‚ow solution. Figures 8.4(a) and 8.4.2(b) show the active power dominions, and
Figure 8.5(a) and 8.5(b) show the reactive power dominions.




40 + j5
45 + j15
0.29
80.15


Lake Main
North 14.87
32.94 32.23 14.89


47.20 1.08 1.37 2.02
5.20 4.29
5.01 2.16
30.14
27.66
4.85
5.50
5.15
46.84
30.61
28.06
1.96 5.00 3.56
1.53


Elm
South 56.06 55.00


20 + j10 60 + j10
6.44
6.07
87.89 14.41

Optimal power ¬‚ow solution
Figure 8.3
317
DOMINIONS




(a) (b)

Figure 8.4 The active dominion of the generators: (a) Gen“North and (b) Gen“South




(a) (b)

The reactive dominion of the generators: (a) Gen“North and (b) Gen“South
Figure 8.5




8.4.1 Dominion Contributions to Active Power Flows

Building on the ideas advanced in Section 8.2, the two-in¬‚ow, two-out¬‚ow system shown in
Figure 8.2 is modi¬ed, as shown in Figure 8.6, to include n in¬‚ows, with one of the out¬‚ows
being a transmission line and the other a load.
The active and reactive power contribution of each dominion or generator to the branch
and load is determined by using the proportional sharing principle demonstrated in
Section 8.3. In this section the issue of active power is addressed, and Figure 8.6 re¬‚ects this
point.
The power ¬‚ow at the sending end of line m is made up of the contribution of the n
in¬‚ows and the generator. Similarly, the load PL is fed by the contribution of the n in¬‚ows
and the generator.
Expanding on the result given in Equation (8.14) to encompass n in¬‚ows but restricted to
branch mk (the load will be treated separately in Section 8.4.3), the following equations
318 POWER FLOW TRACING



PD1


P ′mk P′′mk
PD2
¦


k
PD n


PG

PL
m
Contributions of active power dominions to branch mk
Figure 8.6



apply at the sending end of the branch:

P0mk ¼ P0D1 þ P0D2 þ Á Á Á þ P0Dn þ P0G ; °8:18Þ

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