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Page 1
5 TRANSMISSION LOSS
Generally, in a power system, several plants are situated at different
places. They are interconnected by long transmission lines. The entire
system load along with transmission loss shall be met by the power plants
in the system. Transmission loss depends on i) line parameters ii) bus
voltages and iii) power flow. Determination of transmission loss requires
complex computations. However, with reasonable approximations, for a
power system with N number of power plants, transmission loss can be
represented as
[ ]
N 2 1 L
P P P P … =
?
?
?
?
?
?
?
?
?
?
?
?
NN N2 N1
2N 22 21
1N 12 11
B B B
B B B
B B B
?
?
?
?
?
?
?
?
?
?
?
?
N
2
1
P
P
P
(15)
where
N 2 1
P ,......., P , P are the powers supplied by the plants 1,2,…….,N
respectively.
Page 2
5 TRANSMISSION LOSS
Generally, in a power system, several plants are situated at different
places. They are interconnected by long transmission lines. The entire
system load along with transmission loss shall be met by the power plants
in the system. Transmission loss depends on i) line parameters ii) bus
voltages and iii) power flow. Determination of transmission loss requires
complex computations. However, with reasonable approximations, for a
power system with N number of power plants, transmission loss can be
represented as
[ ]
N 2 1 L
P P P P … =
?
?
?
?
?
?
?
?
?
?
?
?
NN N2 N1
2N 22 21
1N 12 11
B B B
B B B
B B B
?
?
?
?
?
?
?
?
?
?
?
?
N
2
1
P
P
P
(15)
where
N 2 1
P ,......., P , P are the powers supplied by the plants 1,2,…….,N
respectively.
The coefficient matrix [ B ] is called as LOSS COEFFICIENT MATRIX or
B
mn
COEFFICIENT MATRIX or B – COEFFICIENT MATRIX. This matrix is
symmetric.
From eq.(15) P
L
= [ ]
N 2 1
P P P … …
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
=
=
N
1 n
n n N
N
1 n
n n 2
N
1 n
n n 1
P B
P B
P B
= P
N
P B
?
+ P
N
P B
?
+ ……. + P
N
P B
?
= P
1
n
1 n
n 1
P B
?
=
+ P
2
n
1 n
n 2
P B
?
=
+ ……. + P
N
n
1 n
n N
P B
?
=
=
n
N
1 n
n 1 1
P B P
?
=
+
n
N
1 n
n 2 2
P B P
?
=
+ …… +
n
N
1 n
n N N
P B P
?
=
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
Thus P
L
can be written as
P
L
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
(16)
Page 3
5 TRANSMISSION LOSS
Generally, in a power system, several plants are situated at different
places. They are interconnected by long transmission lines. The entire
system load along with transmission loss shall be met by the power plants
in the system. Transmission loss depends on i) line parameters ii) bus
voltages and iii) power flow. Determination of transmission loss requires
complex computations. However, with reasonable approximations, for a
power system with N number of power plants, transmission loss can be
represented as
[ ]
N 2 1 L
P P P P … =
?
?
?
?
?
?
?
?
?
?
?
?
NN N2 N1
2N 22 21
1N 12 11
B B B
B B B
B B B
?
?
?
?
?
?
?
?
?
?
?
?
N
2
1
P
P
P
(15)
where
N 2 1
P ,......., P , P are the powers supplied by the plants 1,2,…….,N
respectively.
The coefficient matrix [ B ] is called as LOSS COEFFICIENT MATRIX or
B
mn
COEFFICIENT MATRIX or B – COEFFICIENT MATRIX. This matrix is
symmetric.
From eq.(15) P
L
= [ ]
N 2 1
P P P … …
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
=
=
N
1 n
n n N
N
1 n
n n 2
N
1 n
n n 1
P B
P B
P B
= P
N
P B
?
+ P
N
P B
?
+ ……. + P
N
P B
?
= P
1
n
1 n
n 1
P B
?
=
+ P
2
n
1 n
n 2
P B
?
=
+ ……. + P
N
n
1 n
n N
P B
?
=
=
n
N
1 n
n 1 1
P B P
?
=
+
n
N
1 n
n 2 2
P B P
?
=
+ …… +
n
N
1 n
n N N
P B P
?
=
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
Thus P
L
can be written as
P
L
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
(16)
P
L
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
(16)
When the powers are in MW, the B
mn
coefficients are of dimension 1/ MW.
If powers are in per-unit, then B
mn
coefficients are also in per-unit. Loss
coefficient matrix of a power system shall be determined before hand and
made available for economic dispatch.
For a two plant system, the expression for the transmission loss is For a two plant system, the expression for the transmission loss is
[ ]
?
?
?
?
?
?
?
?
?
?
?
?
=
2
1
22 21
12 11
2 1 L
P
P
B B
B B
P P P
[ ]
2 1
P P =
?
?
?
?
?
?
+
+
2 22 1 21
2 12 1 11
P B P B
P B P B
=
22
2
2 21 2 1 12 2 1 11
2
1
B P B P P B P P B P + + +
Page 4
5 TRANSMISSION LOSS
Generally, in a power system, several plants are situated at different
places. They are interconnected by long transmission lines. The entire
system load along with transmission loss shall be met by the power plants
in the system. Transmission loss depends on i) line parameters ii) bus
voltages and iii) power flow. Determination of transmission loss requires
complex computations. However, with reasonable approximations, for a
power system with N number of power plants, transmission loss can be
represented as
[ ]
N 2 1 L
P P P P … =
?
?
?
?
?
?
?
?
?
?
?
?
NN N2 N1
2N 22 21
1N 12 11
B B B
B B B
B B B
?
?
?
?
?
?
?
?
?
?
?
?
N
2
1
P
P
P
(15)
where
N 2 1
P ,......., P , P are the powers supplied by the plants 1,2,…….,N
respectively.
The coefficient matrix [ B ] is called as LOSS COEFFICIENT MATRIX or
B
mn
COEFFICIENT MATRIX or B – COEFFICIENT MATRIX. This matrix is
symmetric.
From eq.(15) P
L
= [ ]
N 2 1
P P P … …
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
=
=
N
1 n
n n N
N
1 n
n n 2
N
1 n
n n 1
P B
P B
P B
= P
N
P B
?
+ P
N
P B
?
+ ……. + P
N
P B
?
= P
1
n
1 n
n 1
P B
?
=
+ P
2
n
1 n
n 2
P B
?
=
+ ……. + P
N
n
1 n
n N
P B
?
=
=
n
N
1 n
n 1 1
P B P
?
=
+
n
N
1 n
n 2 2
P B P
?
=
+ …… +
n
N
1 n
n N N
P B P
?
=
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
Thus P
L
can be written as
P
L
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
(16)
P
L
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
(16)
When the powers are in MW, the B
mn
coefficients are of dimension 1/ MW.
If powers are in per-unit, then B
mn
coefficients are also in per-unit. Loss
coefficient matrix of a power system shall be determined before hand and
made available for economic dispatch.
For a two plant system, the expression for the transmission loss is For a two plant system, the expression for the transmission loss is
[ ]
?
?
?
?
?
?
?
?
?
?
?
?
=
2
1
22 21
12 11
2 1 L
P
P
B B
B B
P P P
[ ]
2 1
P P =
?
?
?
?
?
?
+
+
2 22 1 21
2 12 1 11
P B P B
P B P B
=
22
2
2 21 2 1 12 2 1 11
2
1
B P B P P B P P B P + + +
Since B
mn
coefficient matrix is symmetric, for two plant system
22
2
2 12 2 1 11
2
1 L
B P B P P 2 B P P + + = (17)
In later calculations we need the Incremental Transmission Loss ( ITL ),
i
L
P
P
?
?
.
For two plant system
2 12 1 11
L
P B 2 P B 2
P
P
+ =
?
?
(18)
2 12 1 11
1
P B 2 P B 2
P
+ =
?
(18)
2 22 1 12
2
L
P B 2 P B 2
P
P
+ =
?
?
(19)
This can be generalized as
n
P B 2
P
P
N
1 n
mn
m
L
?
=
=
?
?
m = 1,2,………., N (20)
Page 5
5 TRANSMISSION LOSS
Generally, in a power system, several plants are situated at different
places. They are interconnected by long transmission lines. The entire
system load along with transmission loss shall be met by the power plants
in the system. Transmission loss depends on i) line parameters ii) bus
voltages and iii) power flow. Determination of transmission loss requires
complex computations. However, with reasonable approximations, for a
power system with N number of power plants, transmission loss can be
represented as
[ ]
N 2 1 L
P P P P … =
?
?
?
?
?
?
?
?
?
?
?
?
NN N2 N1
2N 22 21
1N 12 11
B B B
B B B
B B B
?
?
?
?
?
?
?
?
?
?
?
?
N
2
1
P
P
P
(15)
where
N 2 1
P ,......., P , P are the powers supplied by the plants 1,2,…….,N
respectively.
The coefficient matrix [ B ] is called as LOSS COEFFICIENT MATRIX or
B
mn
COEFFICIENT MATRIX or B – COEFFICIENT MATRIX. This matrix is
symmetric.
From eq.(15) P
L
= [ ]
N 2 1
P P P … …
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
=
=
N
1 n
n n N
N
1 n
n n 2
N
1 n
n n 1
P B
P B
P B
= P
N
P B
?
+ P
N
P B
?
+ ……. + P
N
P B
?
= P
1
n
1 n
n 1
P B
?
=
+ P
2
n
1 n
n 2
P B
?
=
+ ……. + P
N
n
1 n
n N
P B
?
=
=
n
N
1 n
n 1 1
P B P
?
=
+
n
N
1 n
n 2 2
P B P
?
=
+ …… +
n
N
1 n
n N N
P B P
?
=
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
Thus P
L
can be written as
P
L
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
(16)
P
L
=
n
P B P
n m
N
1 m
N
1 n
m ? ?
= =
(16)
When the powers are in MW, the B
mn
coefficients are of dimension 1/ MW.
If powers are in per-unit, then B
mn
coefficients are also in per-unit. Loss
coefficient matrix of a power system shall be determined before hand and
made available for economic dispatch.
For a two plant system, the expression for the transmission loss is For a two plant system, the expression for the transmission loss is
[ ]
?
?
?
?
?
?
?
?
?
?
?
?
=
2
1
22 21
12 11
2 1 L
P
P
B B
B B
P P P
[ ]
2 1
P P =
?
?
?
?
?
?
+
+
2 22 1 21
2 12 1 11
P B P B
P B P B
=
22
2
2 21 2 1 12 2 1 11
2
1
B P B P P B P P B P + + +
Since B
mn
coefficient matrix is symmetric, for two plant system
22
2
2 12 2 1 11
2
1 L
B P B P P 2 B P P + + = (17)
In later calculations we need the Incremental Transmission Loss ( ITL ),
i
L
P
P
?
?
.
For two plant system
2 12 1 11
L
P B 2 P B 2
P
P
+ =
?
?
(18)
2 12 1 11
1
P B 2 P B 2
P
+ =
?
(18)
2 22 1 12
2
L
P B 2 P B 2
P
P
+ =
?
?
(19)
This can be generalized as
n
P B 2
P
P
N
1 n
mn
m
L
?
=
=
?
?
m = 1,2,………., N (20)
6 ECONOMIC DIVISION OF SYSTEM LOAD BETWEEN VARIOUS PLANTS
IN THE POWER SYSTEM
It is to be noted that different plants in a power system will have different
cost characteristics.
Consider a power system having N number of plants. Input-output
characteristics of the plants are denoted as ) P ( C ...., ),........ P ( C ), P ( C
N N 2 2 1 1
.
Our problem is for a given system load demand P
D
, find the set of plant
generation
N 2 1
P ., ,......... P , P which minimizes the cost function
C
T
= ) P ( C .. .......... ) P ( C ) P ( C
N N 2 2 1 1
+ + + (21)
subject to the constraints
0 ) P .. .......... P P ( P P
N 2 1 L D
= + + + - + (22)
and
max i i min i
P P P = = I = 1,2,……….,N (23)
Inequality constraints are omitted for the time being.
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Nov 15, 2024 Last updated |
Document Description: Economic Load Dispatch (With Losses) for Electrical Engineering (EE) 2024 is part of Power Systems preparation.
The notes and questions for Economic Load Dispatch (With Losses) have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about Economic Load Dispatch (With Losses) covers topics
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