Formula Sheets: Economic Load Dispatch | Power Systems - Electrical Engineering (EE) PDF Download

 Page 1


Power S ystems F ormula Sheet: Economic
Load Dispatch
1. Economic Load Dispatch (ELD) Basics
• Objective : Minimize total fuel cost of gener ation while meeting load demand
and constr aints.
• Cost Function : F or thei -th gener ator:
F
i
(P
i
) = a
i
P
2
i
+b
i
P
i
+c
i
(Rs/h or $/h)
whereP
i
is power output (MW),a
i
,b
i
,c
i
are cost coefficients.
• Incremental Cost (Fuel cost per unit power):
?
i
=
dF
i
dP
i
= 2a
i
P
i
+b
i
(Rs/MWh or $/MWh)
2. ELD without Tr ansmission Losses
• Power Balance Constr aint :
n
?
i=1
P
i
= P
D
whereP
D
is total load demand,n is number of gener ators.
• Optimization Condition : F or minimum cost, all gener ators oper ate at equal
incremental cost (Coordination Equation):
dF
1
dP
1
=
dF
2
dP
2
=··· =
dF
n
dP
n
= ?
2a
i
P
i
+b
i
= ? for alli
• Solution for Power Output :
P
i
=
?-b
i
2a
i
Substitute into power balance to solve for? :
n
?
i=1
?-b
i
2a
i
= P
D
1
Page 2


Power S ystems F ormula Sheet: Economic
Load Dispatch
1. Economic Load Dispatch (ELD) Basics
• Objective : Minimize total fuel cost of gener ation while meeting load demand
and constr aints.
• Cost Function : F or thei -th gener ator:
F
i
(P
i
) = a
i
P
2
i
+b
i
P
i
+c
i
(Rs/h or $/h)
whereP
i
is power output (MW),a
i
,b
i
,c
i
are cost coefficients.
• Incremental Cost (Fuel cost per unit power):
?
i
=
dF
i
dP
i
= 2a
i
P
i
+b
i
(Rs/MWh or $/MWh)
2. ELD without Tr ansmission Losses
• Power Balance Constr aint :
n
?
i=1
P
i
= P
D
whereP
D
is total load demand,n is number of gener ators.
• Optimization Condition : F or minimum cost, all gener ators oper ate at equal
incremental cost (Coordination Equation):
dF
1
dP
1
=
dF
2
dP
2
=··· =
dF
n
dP
n
= ?
2a
i
P
i
+b
i
= ? for alli
• Solution for Power Output :
P
i
=
?-b
i
2a
i
Substitute into power balance to solve for? :
n
?
i=1
?-b
i
2a
i
= P
D
1
3. ELD with Tr ansmission Losses
• Power Balance with Losses :
n
?
i=1
P
i
= P
D
+P
L
whereP
L
is tr ansmission loss.
• Tr ansmission Loss F ormula :
P
L
=
n
?
i=1
n
?
j=1
P
i
B
ij
P
j
whereB
ij
are loss coefficients (B-matrix).
• Coordination Equation with Losses :
dF
i
dP
i
·
1
1-
?P
L
?P
i
= ?
• Incremental Tr ansmission Loss :
?P
L
?P
i
= 2
n
?
j=1
B
ij
P
j
• Penalty F actor (L
i
):
L
i
=
1
1-
?P
L
?P
i
• Optimal Dispatch Condition :
L
i
·
dF
i
dP
i
= ? or L
i
(2a
i
P
i
+b
i
) = ?
4. Constr aints
• Gener ator Limits :
P
i, min
= P
i
= P
i, max
If P
i
violates limits, set P
i
= P
i, min
or P
i, max
and re-solve for remaining gener a-
tors.
• Equality Constr aint : Power balance (with or without losses).
• Inequality Constr aints : Gener ator power limits, line flow limits, etc.
2
Page 3


Power S ystems F ormula Sheet: Economic
Load Dispatch
1. Economic Load Dispatch (ELD) Basics
• Objective : Minimize total fuel cost of gener ation while meeting load demand
and constr aints.
• Cost Function : F or thei -th gener ator:
F
i
(P
i
) = a
i
P
2
i
+b
i
P
i
+c
i
(Rs/h or $/h)
whereP
i
is power output (MW),a
i
,b
i
,c
i
are cost coefficients.
• Incremental Cost (Fuel cost per unit power):
?
i
=
dF
i
dP
i
= 2a
i
P
i
+b
i
(Rs/MWh or $/MWh)
2. ELD without Tr ansmission Losses
• Power Balance Constr aint :
n
?
i=1
P
i
= P
D
whereP
D
is total load demand,n is number of gener ators.
• Optimization Condition : F or minimum cost, all gener ators oper ate at equal
incremental cost (Coordination Equation):
dF
1
dP
1
=
dF
2
dP
2
=··· =
dF
n
dP
n
= ?
2a
i
P
i
+b
i
= ? for alli
• Solution for Power Output :
P
i
=
?-b
i
2a
i
Substitute into power balance to solve for? :
n
?
i=1
?-b
i
2a
i
= P
D
1
3. ELD with Tr ansmission Losses
• Power Balance with Losses :
n
?
i=1
P
i
= P
D
+P
L
whereP
L
is tr ansmission loss.
• Tr ansmission Loss F ormula :
P
L
=
n
?
i=1
n
?
j=1
P
i
B
ij
P
j
whereB
ij
are loss coefficients (B-matrix).
• Coordination Equation with Losses :
dF
i
dP
i
·
1
1-
?P
L
?P
i
= ?
• Incremental Tr ansmission Loss :
?P
L
?P
i
= 2
n
?
j=1
B
ij
P
j
• Penalty F actor (L
i
):
L
i
=
1
1-
?P
L
?P
i
• Optimal Dispatch Condition :
L
i
·
dF
i
dP
i
= ? or L
i
(2a
i
P
i
+b
i
) = ?
4. Constr aints
• Gener ator Limits :
P
i, min
= P
i
= P
i, max
If P
i
violates limits, set P
i
= P
i, min
or P
i, max
and re-solve for remaining gener a-
tors.
• Equality Constr aint : Power balance (with or without losses).
• Inequality Constr aints : Gener ator power limits, line flow limits, etc.
2
5. Lambda Iter ation Method
– Steps :
1. Choose initial? .
2. ComputeP
i
=
?-b
i
2a
i
(without losses) orP
i
=
?/L
i
-b
i
2a
i
(with losses).
3. Check power balance:
?
P
i
= P
D
+P
L
.
4. A djust? :
If
?
P
i
> P
D
+P
L
, decrease?; if
?
P
i
< P
D
+P
L
, increase?.
5. Repeat until power balance is satisfied within toler ance.
6. T otal Cost
*
T otal Fuel Cost :
F
total
=
n
?
i=1
F
i
(P
i
) =
n
?
i=1
(a
i
P
2
i
+b
i
P
i
+c
i
)
3