1. Introductiona
Fig. 1 is a simple four-bus system. Two load buses 3 and 4 are fed by two generators 1 and
2. Line parameters are given in Table 1. Specified bus data(the italicized) and the power- flow solutions for base case are
shown in Table 2(1).
Fig. 1. Single line diagram of four-bus system
Table 1. Line parameters of four-bus system(p.u.)
|
from
|
to
|
R
|
X
|
Shunt Y
|
|
1
|
4
|
.00744
|
.0372
|
0.0775
|
|
1
|
3
|
.01008
|
.0504
|
0.1025
|
|
2
|
3
|
.00744
|
.0372
|
0.0775
|
|
2
|
4
|
.01272
|
.0636
|
0.1275
|
Table 2. Specified bus data and base-case power-flow(p.u.)
|
Bus
|
P
|
Q
|
V
|
Angle(deg)
|
|
1
|
1.913152
|
1.87224
|
1.0
|
0
|
|
2
|
3.18
|
1.32543
|
1.0
|
2.43995
|
|
3
|
-2.20
|
-1.3634
|
.96051
|
-1.0793
|
|
4
|
-2.80
|
-1.7352
|
.94304
|
-2.6265
|
The following is the classical ELD formulation including system loss(2).
where fi, P$_{Gi}$ and PFi are the cost function, MW and the penalty factor defined
for the i-th generator. P$_{loss}$ is the system active power loss.
The power loss sensitivities are directly used in ELD computation as shown in (2). What we need initially for solving (1) and (2) is calculating the active power loss sensitivities(P$_{loss}$ sensitivities from
now on) for all generators including the slack bus - ¶P$_{loss}$/¶P$_{G1}$ and ¶P$_{loss}$/¶P$_{G2}$.
H. Happ presented the following formula for calculating the power loss sensitivities
(3).
The angle reference - the reference for the angles of all bus voltages in the system
- has been specified conventionally on the slack bus as shown in Table 2. But when the angle reference is specified on the slack bus, the loss sensitivity
of the slack bus cannot be derived directly in (3).
The angle reference, however, can be specified on any bus in the system without affecting
the power flow solution(4). The loss sensitivities for all generators including the slack bus can be obtained
by specifying the angle reference on a bus that has no generation(4-6), and the authors call this ‘angle reference transposition’(from now on ART) in this
paper.
In this paper, the reactive power loss sensitivities(from now on Q$_{loss}$ sensitivities)
as well as the P$_{loss}$ sensitivities including the slack bus are derived by ART.
And these loss sensitivities are tested on four cases of power system computation
to optimize the operating cost and system losses.
2. Derivation of loss sensitivities by ART
2.1 ART and re-construction of Jacobian
Let us take a look at the angle terms in the following power flow equations given
for bus k:
As shown in (4), what is actually needed for the bus angle in power flow computation is not the magnitude
itself or the location of the angle reference (qref from now on), but is in fact the
relative angle difference $\theta$$\theta$ km=$\theta$$\theta$ k-qm, where qk and
qm are the voltage angles of bus k and adjacent bus m. Therefore, qref does not necessarily
have to be specified on the very slack bus but it can be assigned on any other bus
in the system as many of the power system analysts already know. For example, we assign
only the magnitude of the voltage on the slack bus(e.g., |V1|=1.02) and then we can
specify qref on any other bus(e.g., $\theta_{4}$=0) without affecting the power flow
solutions(4).
The construction of the Jacobian matrix is changed according to the transposition
of qref. The general form of the mismatch equation with full 8ⅹ8 Jacobian matrix for
a four-bus system is given as follows.
$$\text { where } J = \left[\begin{array}{llllllll}a_{11} & a_{12} & a_{13} & a_{14}
& a_{15} & a_{16} & a_{17} & a_{18} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25}
& a_{26} & a_{27} & a_{28} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36}
& a_{37} & a_{38} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} & a_{47}
& a_{48} \\ a_{51} & a_{52} & a_{83} & a_{54} & a_{55} & a_{56} & a_{57} & a_{58}
\\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} & a_{67} & a_{65} \\ a_{71}
& a_{72} & a_{73} & a_{74} & a_{75} & a_{76} & a_{77} & a_{78} \\ a_{81} & a_{82}
& a_{83} & a_{84} & a_{85} & a_{86} & a_{87} & a_{88}\end{array}\right]
$$
$$
=\left[\begin{array}{llllllll}
\frac{\partial P_{1}}{\partial \theta_{1}} & \frac{\partial P_{1}}{\partial \theta_{2}}
& \frac{\partial P_{1}}{\partial \theta_{3}} & \frac{\partial P_{1}}{\partial \theta_{4}}
& \frac{\partial P_{1}}{\partial V_{1}} & \frac{\partial P_{1}}{\partial V_{2}} &
\frac{\partial P_{1}}{\partial V_{3}} & \frac{\partial P_{1}}{\partial V_{4}} \\
\frac{\partial P_{2}}{\partial \theta_{1}} & \frac{\partial P_{2}}{\partial \theta_{2}}
& \frac{\partial P_{2}}{\partial \theta_{3}} & \frac{\partial P_{2}}{\partial \theta_{4}}
& \frac{\partial P_{2}}{\partial V_{1}} & \frac{\partial P_{2}}{\partial V_{2}} &
\frac{\partial P_{2}}{\partial V_{3}} & \frac{\partial P_{2}}{\partial V_{4}} \\
\frac{\partial P_{3}}{\partial \theta_{1}} & \frac{\partial P_{3}}{\partial \theta_{2}}
& \frac{\partial P_{3}}{\partial \theta_{3}} & \frac{\partial P_{3}}{\partial \theta_{4}}
& \frac{\partial P_{3}}{\partial V_{1}} & \frac{\partial P_{3}}{\partial V_{2}} &
\frac{\partial P_{3}}{\partial V_{3}} & \frac{\partial P_{3}}{\partial V_{4}} \\
\frac{\partial P_{4}}{\partial \theta_{1}} & \frac{\partial P_{4}}{\partial \theta_{2}}
& \frac{\partial P_{4}}{\partial \theta_{3}} & \frac{\partial P_{4}}{\partial \theta_{4}}
& \frac{\partial P_{4}}{\partial V_{1}} & \frac{\partial P_{4}}{\partial V_{2}} &
\frac{\partial P_{4}}{\partial V_{3}} & \frac{\partial P_{4}}{\partial V_{4}} \\
\frac{\partial Q_{1}}{\partial \theta_{1}} & \frac{\partial Q_{1}}{\partial \theta_{2}}
& \frac{\partial Q_{1}}{\partial \theta_{3}} & \frac{\partial Q_{1}}{\partial \theta_{4}}
& \frac{\partial Q_{1}}{\partial V_{1}} & \frac{\partial Q_{1}}{\partial V_{2}} &
\frac{\partial Q_{1}}{\partial V_{3}} & \frac{\partial Q_{1}}{\partial V_{4}} \\
\frac{\partial Q_{2}}{\partial \theta_{1}} & \frac{\partial Q_{2}}{\partial \theta_{2}}
& \frac{\partial Q_{2}}{\partial \theta_{3}} & \frac{\partial Q_{2}}{\partial \theta_{4}}
& \frac{\partial Q_{2}}{\partial V_{1}} & \frac{\partial Q_{2}}{\partial V_{2}} &
\frac{\partial Q_{2}}{\partial V_{3}} & \frac{\partial Q_{2}}{\partial V_{4}} \\
\frac{\partial Q_{3}}{\partial \theta_{1}} & \frac{\partial Q_{3}}{\partial \theta_{2}}
& \frac{\partial Q_{3}}{\partial \theta_{3}} & \frac{\partial Q_{3}}{\partial \theta_{4}}
& \frac{\partial Q_{3}}{\partial V_{1}} & \frac{\partial Q_{3}}{\partial V_{2}} &
\frac{\partial Q_{3}}{\partial V_{3}} & \frac{\partial Q_{3}}{\partial I_{4}} \\
\frac{\partial Q_{1}}{\partial \theta_{1}} & \frac{\partial Q_{4}}{\partial \theta_{2}}
& \frac{\partial Q_{3}}{\partial \theta_{3}} & \frac{\partial Q_{4}}{\partial \theta_{4}}
& \frac{\partial Q_{4}}{\partial V_{1}} & \frac{\partial Q_{1}}{\partial V_{2}} &
\frac{\partial Q_{3}}{\partial V_{3}} & \frac{\partial O_{1}}{\partial V_{4}}
\end{array} \right]
$$
When qref is specified on the slack bus(i.e., setting $\theta_{1}$=0), the elements
related to the slack bus angle are discarded from the Jacobian, and for Fig. 1, equation (5) will be modified with the following new 5ⅹ5 matrix J1 as shown below.
$$
\text { where } J_{1}=\left[\begin{array}{lllll}
\\
& a_{22} & a_{23} & a_{24} & a_{27} & a_{28} \\
& a_{32} & a_{33} & a_{34} & a_{37} & a_{38} \\
& a_{42} & a_{43} & a_{44} & a_{47} & a_{48} \\
& a_{72} & a_{73} & a_{74} & a_{77} & a_{78} \\
& a_{82} & a_{83} & a_{84} & a_{87} & a_{88}
\end{array}\right]
$$
When we move qref onto bus 3(i.e., setting $\theta_{3}$=0), the elements related to
$\theta_{3}$ are discarded from Jacobian, however, there still remain $\theta_{1}$-related
elements in Jacobian. And the mismatch equation will be modified with the following
reconstructed Jacobian J3 as shown below.
$$
\text { where } J_{3}=\left[\begin{array}{lllll}
a_{11} & a_{12} & & a_{14} & a_{17} & a_{18} \\
a_{21} & a_{22} & & a_{24} & a_{27} & a_{28} \\ \\
a_{41} & a_{42} & & a_{44} & a_{47} & a_{48} \\
a_{71} & a_{72} & & a_{74} & a_{77} & a_{78} \\
a_{81} & a_{82} & & a_{84} & a_{87} & a_{88}
\end{array}\right]
$$
The re-construction of the Jacobian does not influence the power flow solution(4).
2.2 Deviation of loss sensitivities by ART
The following is a general form of formula (3) for calculating the power loss sensitivities of system Fig 1.(3)
Remember that J is the same Jacobian matrix in (5) that has been constructed for power flow calculation. Since transposition of qref
changes the Jacobian matrix, it also affects equation (8) that includes Jacobian.
When qref is specified on the slack bus(i.e., setting $\theta_{1}$=0), the elements
related to the slack bus angle are discarded from the Jacobian, and equation (8) will be modified with J1 as shown below. Therefore, the loss sensitivities of the
slack bus cannot be derived directly with (9).
$$
\text { where } J_{1}^{T}=\left[\begin{array}{lllll}
\\
& a_{22} & a_{32} & a_{42} & a_{72} & a_{82} \\
& a_{23} & a_{33} & a_{43} & a_{73} & a_{83} \\
& a_{24} & a_{34} & a_{44} & a_{74} & a_{84} \\
& a_{27} & a_{37} & a_{47} & a_{77} & a_{87} \\
& a_{28} & a_{38} & a_{48} & a_{78} & a_{88}
\end{array}\right]
$$
When qref is moved onto bus 3(i.e., setting $\theta_{3}$=0), the elements related
to $\theta_{3}$ are discarded from Jacobian, however, there remain $\theta_{1}$-related
elements in Jacobian.
Equation (8) will be modified with the reconstructed Jacobian J3, and now we can obtain the loss
sensitivity of the slack bus as shown below(4).
$$
\text { where } J_{3}^{T}=\left[\begin{array}{lllll}
a_{11} & a_{21} & & a_{41} & a_{71} & a_{81} \\
a_{12} & a_{22} & & a_{42} & a_{72} & a_{82} \\ \\
a_{14} & a_{24} & & a_{44} & a_{74} & a_{84} \\
a_{17} & a_{27} & & a_{47} & a_{77} & a_{87} \\
a_{18} & a_{28} & & a_{48} & a_{78} & a_{88}
\end{array}\right]
$$
3. New penalty factor calculation using loss sensitivities derived by ART
ART can be applied to a derivation of new penalty factors. A simple transposition
of the angle reference bus, from the slack bus to another bus where no generation
exists, enables the derivation of the penalty factors for all generators including
slack bus.
For Fig. 1, we can obtain the loss sensitivities, ¶P$_{loss}$/¶P$_{G1}$ and ¶P$_{loss}$/¶P$_{G2}$,
of all generators including the slack bus by placing the angle reference on load bus
3 as shown in (10).
Derived loss sensitivities can be directly substituted into (2) for penalty factor calculation. Equations (1), (2) and (7), (10) are solved simultaneously with power flow equations (4) in order to obtain economic dispatch.
Cost functions fi for generators are assumed(1):
The final ELD computation results using new penalty factors derived by ART are shown
in Table 3 and compared with the existing method based on the following formulation (7).
where Pref and fref are the generation and cost function for the slack bus, and bi
is the ratio of -DPref to DPGi [See Appendix].
In Table 3, ELD computation results with new penalty factors derived by ART show exactly the
same optimal cost and generation allocation as existing method using (12).
We can also obtain the same optimal solutions when another load bus 4 is specified
as the angle reference bus as also shown in Table 3.(4)
Note that in Table 3, though the penalty factors derived are all different each other according to computation
algorithms and locations of qref, the ratios of PF2/PF1 are determined the same 1.01699.
Table 3. Comparison of ELD results using loss sensitivities derived by ART with existing
method in (7)
|
|
Existing method in (7)
|
New method using ART
|
|
For
$\theta_{3}$=0
|
For
$\theta_{4}$=0
|
|
Total cost
($/hour)
|
4557.31
|
4557.31
|
4557.31
|
|
P$_{G1}$ (MW)
|
195.9367
|
195.9367
|
195.9367
|
|
P$_{G2}$ (MW)
|
313.2978
|
313.2978
|
313.2978
|
|
Transmission
Loss (MW)
|
9.23449
|
9.23449
|
9.23449
|
|
¶P$_{loss}$/¶P$_{G1}$
|
-
|
.010867
|
.023511
|
|
¶P$_{loss}$/¶P$_{G2}$
|
-
|
.027392
|
.039824
|
|
Penalty factor
PF$_{1}$
|
(1.0)
|
1.010987
|
1.024077
|
|
Penalty factor
PF$_{2}$
|
1.01699
|
1.028163
|
1.041476
|
|
PF$_{2}$/PF$_{1}$
|
1.01699
|
1.01699
|
1.01699
|
|
Incremental cost ¶f$_{1}$/¶P$_{G1}$
|
9.567493
|
9.567493
|
9.567493
|
|
Incremental cost
¶f$_{2}$/¶P$_{G2}$
|
9.407659
|
9.407659
|
9.407659
|
4. Calculation of optimal P-Q generation for cost minimization using loss sensitivities
derived by ART
Optimal allocation of the reactive power generations can also contribute to cost reduction.
Adding the Mvar balance equation to the classical ELD formulation yields:
where NG : number of generating units,
fi : operating cost for i-th generator,
PD , QD : total MW/Mvar load,
PGi , QGi : MW/Mvar output of i-th generator,
PG , QG : vector of generator MW/Mvar outputs.
Ploss , Qloss : MW/Mvar transmission loss
The optimality conditions can be obtained using the principle of Lagrangian multiplier
as follows(5):
, where $\mu$P and $\mu$Q are the Lagrangian multipliers.
What we need initially to solve (14) is four kinds of loss sensitivities, ¶P$_{loss}$/¶P$_{G}$, ¶P$_{loss}$/¶Q$_{G}$,
¶Q$_{loss}$/¶P$_{G}$ and ¶Q$_{loss}$/¶Q$_{G}$, for all generators including the slack
bus. We can obtain them all by placing the angle reference on the load bus 4 as shown
below(3).
Substituting these loss sensitivities into (14) and solving simultaneously with power flow equations (4), the optimal P-Q generation allocation for minimizing the operating cost can be obtained.
Comparison Study
A simulation has been performed for Fig. 1 and the following three cases are compared with each other in Table 4.
- Case I : Cost calculation by classical ELD computation
based on (1) and (2), in which the magnitudes of
generator voltage V1 and V2 are given 1.0 p.u.
- Case II : Cost calculation by an existing OPF program(8,9).
- Case III : Cost calculation based on (14) using the loss
sensitivities derived by ART
In Case II and III, the generator voltages are set free in order to obtain optimal
Q generations for minimizing the fuel cost.
For fair competition of each cost minimization algorithm, the authors assumed that
the voltage magnitude 0.94306288 p.u. of load bus 4 determined by Case I is maintained
at the same value also for Case II and III. No other operating constraints such as
the upper and lower power limit of generators have been assumed for convenience. The
same generator cost functions as (11) are assumed.
Table 4. Comparison of operating cost, P-Q generations, voltages and loss sensitivities
for V4=0. 94306288 pu.
|
|
Case I
by ELD computation
|
Case II
by existing OPF program
|
Case III
by proposed method
|
|
Total Cost ($/hr)
|
4557.31
|
4556.71
|
4556.71
|
|
P$_{G1}$ - Q$_{G1}$
|
1.959 / 1.863
|
1.951 / 1.599
|
1.951 / 1.599
|
|
P$_{G2}$ - Q$_{G2}$
|
3.132 / 1.330
|
3.140/ 1.590
|
3.140/ 1.590
|
|
V1
|
1.0
|
.9956
|
.9956
|
|
V2
|
1.0
|
1.0075
|
1.0075
|
|
V3
|
.9605
|
.9631
|
.9631
|
|
V4
|
.94306288
|
|
P$_{loss}$
|
.0923
|
.0918
|
.0918
|
|
Loss-Sensitivities of Case III
¶P$_{loss}$/¶P$_{G1}$ =.021 ¶P$_{loss}$/¶P$_{G2}$ =.035 ¶P$_{loss}$/¶Q$_{G1}$ =.014
¶P$_{loss}$/¶Q$_{G2}$ =.014
¶Q$_{loss}$/¶P$_{G1}$=.104 ¶Q$_{loss}$/¶P$_{G2}$=.174 ¶Q$_{loss}$/¶Q$_{G1}$=.060
¶Q$_{loss}$/¶Q$_{G2}$=.055
|
In Table 4, the generation allocation by proposed method demonstrates an improved total cost
in comparison to the conventional ELD. Furthermore, the proposed method yields the
same cost and P-Q generation as those calculated by the existing OPF program.(5)
5. Calculation of optimal P generation for system loss minimization using P-loss sensitivities
derived by ART
The problem of optimal P generations for minimizing the system loss can be formulated
as follows:
The optimality condition can be obtained using the principle of Lagrangian multipliers
as follows(6):
where $\mu$ is the Lagrangian multiplier. equation (18) implies that the system loss is minimized when all generators are being operated
with equal loss sensitivities. Derivation of loss sensitivities for all generators
can give the solution for this problem.
For Fig. 1, we can obtain the loss sensitivities, ¶P$_{loss}$/¶¶P$_{G1}$ and ¶P$_{loss}$/¶P$_{G1}$,
of all generators including the slack bus by placing the angle reference on load bus
3 as shown below.
By substituting these loss sensitivities into (18) and solving with power equations (4), the optimal generation allocation for minimizing the system loss can be obtained.
Comparison Study
A simulation has been performed for Fig. 1 and the following three cases are compared with each other in Table 5.
- Case I : System loss for equal generation P$_{G1}$=P$_{G1}$
- Case II : System loss minimization using the loss
sensitivities obtained by B-matrix
- Case III : System loss minimization based on (18)
using P$_{loss}$ sensitivities derived by ART in (19).
The B- matrix used for Case II is expressed as:(1)
Table 5. Comparison of system loss and generation allocation
|
|
when
P$_{G1}$=P$_{G1}$
|
by given
B-matrix
|
by proposed method
|
|
P loss
|
.08612
|
.09008
|
.08567
|
|
P$_{G1}$
|
2.5431
|
2.1063
|
2.7488
|
|
P G2
|
2.5431
|
2.9838
|
2.3369
|
|
∂P loss / ∂P G1
|
.01788
|
.03577
|
.02035
|
|
∂P loss / ∂P G2
|
.02217
|
.03577
|
.02035
|
In Table 5, the generation allocation based on (18) yields the most noticeable improvement in system loss. The B-matrix used above should
be corrected to conform to the new operating condition when shifts of outputs among
generators occur if more accurate results are required(1). However, the optimal generations can be directly calculated by using the loss sensitivities
derived by ART because these loss sensitivities can reflect the current system operating
conditions.(6)
6. Calculation of optimal P-Q generation for minimizing system loss using P-Q loss
sensitivities derived by ART
The problem of optimal both P and Q generations for minimizing the system loss can
be formulated as follows:
Optimality conditions are obtained as follows:
where λP and λQ are the Lagrangian multipliers.
Let us apply formula (23) to Fig 1.
What we need initially to solve (23) is four kinds of the P$_{loss}$ and Q$_{loss}$ sensitivities - ¶P$_{loss}$/¶P$_{G}$,
¶P$_{loss}$/¶Q$_{G}$, ¶Q$_{loss}$/¶P$_{G}$ and ¶Q$_{loss}$/¶Q$_{G}$ - for all generators
including the slack bus.
With the angle reference specified on load bus 4 where no generation exists, the P$_{loss}$
and Q$_{loss}$ sensitivities for all generators including the slack bus can be obtained
as shown below:
Substituting loss sensitivities (24) and (25) into (23) and solving with power equations (4), the optimal P-Q generation allocation for minimizing the system loss can be obtained.
Comparison Study
A simulation has been performed for Fig. 1 in order to compare the following three cases with each other in Table 6.
- Case I : System loss in case of equal generation
P$_{G1}$=P$_{G1}$
- Case II : System loss minimization by (18) that uses
only P$_{loss}$ sensitivities derived by ART.
The results are the same as shown in Table 5 of
Section V.
- Case III : System loss minimization based on (23)
using both P$_{loss}$ and Q$_{loss}$ sensitivities of (24) and (25). Note that the generator voltages V1 and V2 are given 1.0 p.u. in Case I and II,
while they are floated in Case III.
In Case III, the authors intentionally fixed the voltage of bus 4 at the same value
0.9432651 p.u. that is determined in Case I and II in order to compare Case III with
Case II.
Case III - the P-Q generation allocation by (23) - yields smaller system loss compared to Cases I and II as shown in Table 6.
7. Conclusions
Loss sensitivities of all generators including the slack bus can be obtained by specifying
the angle reference on a load bus that has no generation, using the nature of the
Jacobian matrix which is re-constructed by transposition of the angle reference bus.
Table 6. Comparison of system loss, P-Q generations, voltages (p.u) and loss sensitivities
|
|
Case I
|
Case II
|
Case III
|
|
System loss
|
.0861211
|
.08567102
|
.085659136
|
|
P$_{G1}$ / Q$_{G1}$
|
2.543/ 1.761
|
2.748 / 1.730
|
2.739 /1.694
|
|
P G2 / Q$_{G2}$
|
2.543/ 1.400
|
2.336 / 1.429
|
2.345 /1.465
|
|
V1
|
1.0
|
1.0
|
.9994
|
|
V2
|
1.0
|
1.0
|
1.0011
|
|
V3
|
.9607
|
.9607
|
.9611
|
|
V4
|
.9432651
|
|
Loss sensitivities of Case III :
¶P$_{loss}$/¶P$_{G1}$=.02630, ¶P$_{loss}$/¶P$_{G1}$=.02631, P$_{loss}$/¶Q$_{G1}$=.01432,
¶P$_{loss}$/¶Q$_{G2}$=.01439, ¶Q$_{loss}$/¶P$_{G1}$=.12997, ¶Q$_{loss}$/¶P$_{G1}$=.12934,
¶Q$_{loss}$/¶Q$_{G1}$=.06029, ¶Q$_{loss}$/¶Q$_{G2}$ =.05599
|
In this paper, the following four applications to power system computations have been
demonstrated using these loss sensitivities.
• Application 1 : New calculation of penalty factors for
classical ELD computation
• Application 2 : Optimal P-Q generation for cost minimization
• Application 3 : Optimal P generation for system loss
minimization
• Application 4 : Optimal P-Q generation for system loss
minimization
Each application has been tested and verified on a sample system. Simulation results
showed that the loss sensitivities derived by the angle reference transposition can
be a useful tool for obtaining the optimal solutions in power system computations.
The authors expect that the angle reference transposition can be applied to many other
uses in power system computation.
APPENDIX
Calculation of Penalty Factors using Reference Bus Penalty Factor(7)
In a power system with several generator buses and a reference-generator bus(The slack
bus is called ‘reference bus’ in (7)), suppose we change the generation on bus i by DPi , where i=2,3,4,…. We assume that
to compensate for the increase in DPi the generation on reference bus just drops off
by DPref. If nothing else changed, DPref would be the negative of DPi plus the increment
of system loss, that is,
The following equation is obtained when all generators are in economic dispatch.
where fi, fref and Pi, Pref are the cost functions and generator outputs for i-th
generator and the reference bus. bi is calculated by following equation.
Since bi of slack bus is not included in (A4), the penalty factor of the slack bus is not directly calculated but is determined
at the value of 1.0 p.u. as the result of ELD computation in (A3). Penalty factors are calculated without direct derivation of loss sensitivity of
the slack bus. Nevertheless above method is a good method, by which the optimal solution
can be obtained as shown in Table 3.
References
J. Grainger, William D. Stevenson, Jr., 1994, Power System Analysis, Mcgraw Hill
Inc., pp. 548-560
H. H. Happ, 1977, Optimal Power Dispatch-A Comprehensive Survey, IEEE Transaction
on PAS, Vol. 96, No. 3, pp. 841-854
H. H. Happ, 1980, Piecewise Methods and Applications to Power Systems, John Wiley
& Sons Inc., New York, pp. 293-297
S. J. Lee, K. Kim, Feb 2002, Re-construction of Jacobian Matrix by Angle Reference
Transposition and Application to New Penalty Factor Calculation, IEEE Power Engineering
Review, Vol. 22, No. 2, pp. 47-50
S. J. Lee, S. D. Yang, Feb 2006, Derivation of P-Q Loss Sensitivities by Angle Reference
Transposition and An Application to Optimal P-Q Generation for Minimum Cost, IEEE
Trans on Power Sys, Vol. 21, No. 1, pp. 428-430
S. J. Lee, Aug 2003, Calculation of Optimal Generation for System Loss Minimization
Using Loss Sensitivities Derived by Angle Reference Transposition, IEEE Trans on Power
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저자소개
He has worked for S-D E&GC Co., Ltd, for 12 years since 2002 and used to be the Chief
Executive of R&D Center.
He has been a professor of Chuncheon Campus of Korea Poly- technic University since
2014. His research interest includes Power system optimization, Quiescent power cut-off
and Human electric shock.
He published many papers on ELCB (Earth Leakage Circuit-Breakers), Human body protection
against electric shock, Improvement of SPD, Quiescent power cut-off, and etc.
E-mail: cjfwnxkq@hanmail.net
He proposed ‘Angle reference transposition in power flow computation’ on IEEE Power
Engineering Review in 2002, which describes that the loss sensitivities for all generators
including the slack bus can be derived by specific assignment of the angle reference
on a bus where no generation exists, while the angle reference has been specified
conventionally on the slack bus.
He applied these loss sensitivities derived by ‘Angle reference transposition’ to
‘Penalty factor calculation in ELD computation’ [IEEE Power Engineering Review 2002],
‘Optimal MW generation for system loss minimization’ [IEEE Trans 2003, 2006] and etc.
He worked for Korea Electric Power Corporation (KEPCO) for 22 years since 1976, mostly
at Power System Research Center.
He has been a professor of Seoul National University of Science and Technology since
1998.
His research interest includes power generation, large power system and engineering
mathematics.
He received a Ph.D. degree at Chungnam National University in 1995.
E-mail : 85sjlee@seoultech.ac.kr