EEL 6266
Power System Operation and Control
Chapter 4
Transmission System Effects
Transmission Losses
An illustration using a simple system Ploss = 0.0002 P12
consider a two generator system P1
Min = 70 MW
the generating units are identical Max = 400 MW 500 MW
production costs are modeled P2
using a quadratic equation Min = 70 MW
Max = 400 MW
F1 (P1 ) = F2 (P2 ) = Fi (Pi ) = 400 + 7 Pi + 0.002 Pi 2
the losses on the transmission line are proportional to the square
of the power flow
let both units be loaded to 250 MW Ploss = 12.5 MW
the load would be under served
P1
by 12.5 MW 250 MW
487.5
P2 MW
250 MW
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 2
Transmission Losses
solution using the Lagrange equation
L = F1 (P1 ) + F2 (P2 ) + λ (500 + Ploss − P1 − P2 )
Ploss = 0.0002 P12
then
∂L
= 7.0 + 0.004 P1 − λ (1 − 0.0004 P1 ) = 0
∂P1
∂L
= 7.0 + 0.004 P2 − λ = 0
∂P2
∂L
= 500 + 0.0002 P12 − P1 − P2 = 0
∂λ
solution: P1 = 178.88, P2 = 327.50, Ploss = 6.38 MW
cost = F1(P1) + F2(P2) = 4623.15
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 3
Transmission Losses
if the optimal dispatch is ignored Ploss = 13.93 MW
and generator 1 is set to supply all
P1
the losses, then 263.93 MW
P1 = 263.93 and Ploss = 13.93 MW 500 MW
P2
total cost = F1(263.93) + F2(250) 250 MW
= 4661.84
optimum dispatch tends toward supplying the losses from the unit
close to the load, resulting in a lower value of losses
the best economics are not necessarily attained at minimum
losses P = 2.08 MW loss
the minimum loss solution:
P1 = 102.08 and P2 = 400 MW P1
102.08 MW
Ploss = 2.08 MW 500 MW
total cost = 4655.43 P2
400 MW (at limit)
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 4
Transmission Losses
Derivation of the penalty factor from incremental losses
start with the Lagrange equation for the economic dispatch
N
N
L = ∑ Fi (Pi ) + λ Pload + Ploss (P1 , P2 ,K, PN ) − ∑ Pi
i =1 i =1
∂L
min L → = 0 ∀Pi min ≤ Pi ≤ Pi max
P ∀i =1KN
i ∂Pi
then
∂L dFi (Pi ) ∂P
= − λ 1 − loss = 0
∂Pi dPi ∂Pi
rearranging the equation
−1
∂Ploss dFi (Pi )
1 − ∂P =λ
i dPi
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 5
Transmission Losses
the incremental loss for bus i is defined as
∂Ploss
∂Pi
the penalty factor for bus i is given as
−1
∂Ploss
Pf i = 1 −
∂Pi
if the losses increase for an increase in power from bus i, the
incremental cost is positive and the penalty factor is greater than
unity
the rearranged minimizing equations become
dFi (Pi )
Pf i = λ ∀Pi min ≤ Pi ≤ Pi max
dPi
which are called the coordination equations
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 6
Transmission Losses
Behaviors of the penalty factor on the coordination equations
when transmission losses are ignored, the penalty factor takes
on the value of unity, making the coordination equation the
same as the incremental cost equation
a penalty factor greater than one (Pfi > 1), representing
increased losses for increased generation at bus i, acts on the
coordination equation as if the original incremental cost
function has been slightly increased
graphically being moved upward
a Pfi < 1 acts on the coordination equation as if the incremental
cost has been slightly decreased
graphically being moved downward
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 7
Transmission Losses
Graphical comparison of the coordination equations with
and without accounting for the losses
dF1 dF dF2 dF dF3 dF3
dP1 dF1 Pf1 1 dP2 dF2 Pf 2 2 dP3 dF3 Pf 3
dP1 dP2 dP3
with
dP1 dP2 dP3
penalty
λ ′′ factors
λ′
no penalty
factors
P1′′ P1′ P1 P2′ P2′′ P2 P3′ P3′′ P3
Pf1 = 1.05 Pf2 = 1.00 Pf3 = 0.95
Pi′ = Dispatch ignoring losses
Pi′′= Dispatch with penalty factors
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 8
Transmission Losses
The B matrix loss formula
simplified, practical method for loss and incremental cost
calculations
basic formula
Ploss = PT [B ]P + B0T P + B00
where P is the vector of all generator bus net power injections
[B] is a square loss factor matrix of the same dimension as P
B0 is a loss factor vector of the same length as P
B00 is a loss factor constant
alternative form of the equation
Ploss = ∑∑ Pi Bij Pj + ∑ Bi 0 Pi + B00
i j i
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 9
Transmission Losses
using the B coefficients in the start
select starting value given: total
of Pi, i = 1…N load, Pload
economic dispatch equations
calculate Ploss using B matrix Economic
equality constraint find demand PD = Pload + Ploss Dispatch
N N N N with
φ = −∑ Pi +Pload +∑∑ Pi Bij Pj +∑ Bi 0 Pi +B00 calculate penalty factor eqs. updated
i =1 i =1 j =1 i =1 for Pfi for i = 1…N penalty
factors
incremental cost equations pick starting λ
∂L dFi N
= − λ 1 − 2∑ Bij Pj − Bi 0
solve coordination equations
∂Pi dPi j =1 for Pi for i=1…N
the presence of the incremental
check demand
losses couples together the adjust λ
|ΣPi–PD|<ε?
coordination equations
compare new Pi to Pi of last
makes the solution process iteration, save max. change
more difficult check solution
print max|Pi–Pi′|<δ?
iterative solution algorithm end results
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 10
Transmission Losses
Example
loss coefficients
0.0676 0.00953 − 0.00507 − 0.0766
[ B ] = 0.00953 0.0521 0.00901 B0 = − 0.00342
− 0.00507 0.00901 0.0294 0.0189
B00 = 0.040357
cost functions
F1 (P1 ) = 213.1 + 11.669 P1 + 0.00533P12 50.0 ≤ P1 ≤ 200
F2 (P2 ) = 200.0 + 10.333P2 + 0.00889 P22 37.5 ≤ P2 ≤ 150
F3 (P3 ) = 240.0 + 10.833P3 + 0.00741P32 45.0 ≤ P3 ≤ 180
Pload = 210 MW
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 11
Transmission Losses
Example
iteration results
iteration Ploss PD λ P1 P2 P3
1 17.8 227.8 12.8019 50.00 85.34 92.49
2 11.4 221.4 12.7929 74.59 71.15 75.69
3 9.0 219.0 12.8098 73.47 70.14 75.39
4 8.8 218.8 12.8156 73.67 69.98 75.18
5 8.8 218.8 12.8189 73.65 69.98 75.18
6 8.8 218.8 12.8206 73.65 69.98 75.18
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 12
Transmission Losses
Reference bus versus load center based penalty factors
B matrix assumption
all load currents conform to an equivalent total load current
equivalent load current is the negative sum of all generation
important concepts
an incremental loss is the change in losses ∂Ploss
for an incremental change in generation output ∆P = ∆Pi
∂Pi
loss
the incremental loss for generator bus i assumes
∆Pj = 0 ∀ j ≠ i
that all other generator outputs remain fixed
implied principles when using the B matrix
an incremental increase in generator output is
matched by an equivalent increment in load
alternative approach is to use a reference generator bus
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 13
Transmission Losses
Reference generator bus based penalty factors
reference bus always moves for a change in generation
load demand stays constant
Pi new = Pi old + ∆Pi
Prefnew = Prefold + ∆Pref
a change in reference generation would be the negative of the
generation change
flows in the system changes as a result of any generator
adjustments
the change in flows is apt to cause a change in losses
∆Pref = − ∆Pi + ∆Ploss
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 14
Transmission Losses
Beta factors
ratio of the negative power change in the reference bus to a
change in generator i:
∆Pref = − β i ∆Pi βi = −
∆Pref
=
(∆Pi − ∆Ploss ) = 1 − ∂Ploss
∆Pi ∆Pi ∂Pi
economic dispatch can now be defined as follows
economic dispatch is reached when an incremental power shift
from any generator to the reference results in no change in net
production cost for any arbitrarily small power change
∂Fi (Pi )
C = ∑ Fi (Pi ) ∆C = ∑ ∆Pi = 0 ∆Pi ≤ ε
∂Pi
this implies
dFi (Pi ) dFref (Pref ) dFi (Pi ) dFref (Pref )
∆Ci = ∆Pi + ∆Pref = ∆Pi − β i ∆Pi = 0
dPi dPref dPi dPref
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 15
Transmission Losses
Beta factors
economic dispatch using beta factors
rewriting the equations
dFi (Pi ) dFref (Pref ) 1 dFi (Pi ) dFref (Pref )
∆Pi = β i ∆Pi → ∆Pi = ∆Pi
dPi dPref β i dPi dPref
this is very similar to the coordination equation where the
reciprocal of beta replaces the penalty factor
first order gradient solution method
pick a generation value for the reference bus
set all other generation according to the equation above
dFref (Pref )
check for total demand
dFi (Pi )
readjust the reference as ∆C = ∑ − βi ∆Pi
i ≠ ref
dPi dPref
needed until a solution is
reached
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 16
Transmission Losses
Finding the reference-bus-based penalty factors
computed directly from the N-R power flow equations
seek to find the ratio of power change at the reference bus when
there is a change, ∆Pi at the i-th generator
∂Pref ∂Pref ∂Pref ∂θ i ∂Pref ∂ Ei
∆Pref = ∑ ∆θ i + ∑ ∆ Ei = ∑ ∆Pi + ∑ ∆Pi
i ∂θ i i ∂ Ei i ∂θ i ∂Pi i ∂ Ei ∂Pi
likewise, find the ratio when there is a reactive power change,
∆Qi at the i-th generator
∂Pref ∂Pref ∂Pref ∂θ i ∂Pref ∂ Ei
∆Pref = ∑ ∆θ i + ∑ ∆ Ei = ∑ ∆Qi + ∑ ∆Qi
i ∂θ i i ∂ Ei i ∂θ i ∂Qi i ∂ Ei ∂Qi
the terms ∂Pref∂θi and ∂Pref∂Ei are derived by
differentiating the power equation for the reference bus
© 2002, 2004 Florida State University EEL 6266 Power System Operation and Control 17
Transmission Losses
the remaining terms are taken from the inverse Jacobian matrix
the resulting equation is
∂Pref ∂Pref ∂Pref ∂Pref ∂Pref ∂Pref
∂P L
1 ∂Q1 ∂P2 ∂Q2 ∂PN ∂QN
∂Pref ∂Pref ∂Pref ∂Pref ∂Pref ∂Pref −1
= L [J ]
∂θ1 ∂ E1 ∂θ 2 ∂ E2 ∂θ N ∂ EN
or
∂Pref ∂P1 ∂Pref ∂θ1
∂P ∂Q ∂P ∂ E
ref 1 ref 1 Note: in practice, Gaussian
∂Pref ∂P2 ∂Pref ∂θ 2
[ ]
−1 elimination is employed to
∂P ref ∂Q 2 = J T
∂Pref ∂ E2 find the reference-bus penalty
M M factors.
∂ P
ref ∂ PN ∂Pref ∂θ N
∂Pref ∂QN ∂Pref ∂ E N
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