EE531

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EE531

• Optimization II

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Applications --Economic Dispatch
N generating units are up and synchronized
to serve a load PL The fuel and OM cost in /hr
for unit n to generate power Pn has the form Cn
an bn Pn cn Pn2 The total cost is C?
Cn Determine the generation schedule, P1, P2,
Pn, to supply load at minimum cost
P1
Cn /h
Pn
PL
Pn
Min C? Cn L ? Cn - ? ( P1P2Pn PL)
P1P2Pn PL ?Cn / ?Pn ?
P1P2Pn PL
Units must run at equal Incremental cost
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Applications --Economic Dispatch
P1
N generating units are up and synchronized
to serve a load PL Incremental cost dCn /d Pn -
/MWH cost of next MW
Pn
PL
Cn /h
If dCk /dPk gt dCkj / dPj move 1 MW from unit k
to j - Cost decreases dCk /dPk decreases and
dCkj / dPj increases
Pn
Units must run at equal Incremental cost
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Applications --Economic Dispatch
Find P1, P2, P3. i.e., generation schedule to
minimize fuel cost to serve a given load
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Applications Economic DIspatch
When cost curve is quadratic one way to solve
is The stationarity condition can be written as
C7616 /hr
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Applications Economic DIspatch
P1
PL952 MW C7616 /hr
Pn
PL
Lagrange Multiplier measures the sensitivity of
cost to change in constraint P1P2PnPL
i.e., change in load PL ? is the cost of serving
1 additional MW of load system incremental
cost If load goes up 1 MW we can serve it from
any unit since all have incremental cost of ? If
we want to purchase power the cost should be lt
? If we want to sell the price to charge is gt ?
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Applications Economic Dispatch
P1
How much can we sell if a buyer is willing to pay
15/MWH As we sell more our effective load and ?
go up. So solve
Pn
PL
Sale
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Sale 68 MW Old cost 7616 New cost
8670 /hr Difference 1054 Revenue from sale
1088 /hr Profit 34/hr
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Applications Optimal Power Flow
Extend Economic dispatch to include network
Min f(x,u) generator cost G(x,u) 0
power flow equation H(x,u) limits
u control variables Generator MW, Voltage
Magnitude(Taps, Loads.. x state variables Load
voltage, angles x,u must satisfy power flow
equations Other dependent variables e.g.,
voltage magnitude, line MVA are expressed as
H(x,u) and are subject to limits.
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Applications Optimal Power Flow
Extend Economic dispatch to include network
u control variables P2 and P3 - V1, V2 and
V3 are also controls but are fixed at their
given values in the text example, for
simplicity x state variables ?2 and ?3 - bus
1 is slack so ?10 f cost C1(P1)C2(P2)C3(P3
) - P1 the slack bus power is a dependent
variable g power flow equations that x and u
must satisfy h ignored in this example for
simplicity
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Applications Optimal Power Flow
Extend Economic dispatch to include network
u control variables P2 and P3 x state variables
?2 and ?3 f cost C1(P1)C2(P2)C3(P3)
g power flow equations
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Applications Optimal Power Flow
Extend Economic dispatch to include network
k0 u uo Solve g(x,u) 0 to find xk Find
?uC Stop if ?uC0 Update uk1uk-? ?uC
kk1
Min f(x,u) generator cost G(x,u) 0
power flow equation H(x,u) limits
Dommel Tinney Original Reduced Gradient
Method ?uC ?uf- ?u g T ?x g -T ?xf
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Applications Optimal Power Flow
Extend Economic dispatch to include network
k0 u uo Solve g(x,u) 0 to find xk Find
?uC Stop if ?uC0 Update uk1uk-? ?uC
kk1
Given u solve for x g(x,u) 0 power flow
equation Standard power flow solution xk
xk-1 J-1 (-g(xk,u)
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Applications Optimal Power Flow
Extend Economic dispatch to include network
k0 u uo Solve g(x,u) 0 to find xk Find
?uC Stop if ?uC0 Update uk1uk-? ?uC
kk1
Reduced Gradient ?uC ?uf- ?u g T ?x g -T
?xf
f cost C1(P1)C2(P2)C3(P3) u P2 P3T
generation ?C2/ ?P2 ?uf
?C3/ ?P3
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Applications Optimal Power Flow
Extend Economic dispatch to include network
k0 u uo Solve g(x,u) 0 to find xk Find
?uC Stop if ?uC0 Update uk1uk-? ?uC
kk1
Reduced Gradient ?uC ?uf- ?u g T ?x g -T
?xf
f cost C1(P1)C2(P2)C3(P3) x ?2 ?3T
state ?P1/ ??2 ?xf
?C1/ ?P1 ?C1/ ?P1 ?P1/ ??3
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Applications Optimal Power Flow
Extend Economic dispatch to include network
k0 u uo Solve g(x,u) 0 to find xk Find
?uC Stop if ?uC0 Update uk1uk-? ?uC
kk1
Reduced Gradient ?uC ?uf- ?u g T ?x g -T
?xf
?u g T 1 1 uP1 P2T ?x g
J x ?2 ?3T
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Applications Optimal Power Flow
Extend Economic dispatch to include network
k0 u uo Solve g(x,u) 0 to find xk Find
?uC Stop if ?uC0 Update uk1uk-? ?uC
kk1
Reduced Gradient ?uC ?uf- ?u g T ?x g -T
?xf
uP1 P2T x ?2 ?3T
K0 P20.56 P30.28 Power flow
solution gives ?20.0286 ?2 -0.0186
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Applications Optimal Power Flow
Extend Economic dispatch to include network
k0 u uo Solve g(x,u) 0 to find xk Find
?uC Stop if ?uC0 Update uk1uk-? ?uC
kk1
Reduced Gradient ?uC ?uf- ?u g T ?x g -T
?xf
uP1 P2T x ?2 ?3T
K0
P10.5603 P20.2805
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Applications Optimal Power Flow
Extend Economic dispatch to include network
k0 u uo Solve g(x,u) 0 to find xk Find
?uC Stop if ?uC0 Update uk1uk-? ?uC
kk1
Reduced Gradient ?uC ?uf- ?u g T ?x g -T
?xf
uP1 P2T x ?2 ?3T
K1 P20.5603 P30.2805 Power flow
solution gives ..
Loss 3.3 MW
ED
OPF
7616 /MWH
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Applications Optimal Power Flow
Penalty functions
Min f(x) pTh(x) generator cost inequality
constraint penalty g(x) 0 power
flow equation pk 0 if h(x)0
Large number otherwise
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Applications Optimal Power Flow
LP
Min f(x) Linearize min CTx g(x) 0
Axb h(x)0 Dx 0
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Applications Optimal Power Flow
Other approaches
Min f(x) generator cost g(x) 0
power flow equation h(x)0 limits
L f(x) ?T g(x)µTh(x) ? f ?
g ? ? g µ 0 g(x)0
µTh(x) 0
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Applications Optimal Power Flow
Second order
?L ? f ? g ? ? h µ 0
g(x)0 µTh(x) 0 NR solution
?2f ?2g J ?x - ?L J 0

Put binding constraint hk into g Trick is how to
determine priori if hk is binding
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Applications Security Constrained Optimal Power
Flow
Needed to operate Transmission in Restructured
Energy Markets Lagrange multipliers
give Locational marginal Price
Min f(x) T g(x)0 Base Case
µTh(x) 0 g1(x)0 Contingency 1 (
Line k out) µ1Th1(x) 0 . .
gn(x)0 Contingency 1 ( Line k out)
µnh1(x) 0
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Applications State Estimation
Power system operation and control is based on
real-time measurements Supervisory Control and
Data Acquisitions systems (SCADA) provide
measurements of voltage magnitudes , current,
real and reactive power flow SCADA also provides
current network topology Measurements are
subject to RANDOM, SYSTEMATIC and GROSS
errors Measurements can be used to produce an
estimate of state -State Voltage magnitude
Phase angle - estimated state maximize
likelihood of measurements - Measurements with
Gross errors can be identified and corrected -
all desired quantities can be calculated from
state A data scrubbing process.
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Applications - State Estimation
In the system above State V1, V2 -
Knowing V1 and V2 we can calculate all other
Measurements z1,z2,z3,z4 - Two
voltages and two currents - No direct
measurement of V1 and V2 - Objective estimate
V1 and V2
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Applications - State Estimation

• Objective estimate V1 and V2
• There is a linear relationship between
  measurements and state
• z A x Measurement model
• overdetermined z is 4x1 x is 2x
• -In the absence of noise(errors) mesurements
  would be consistent
• - use two independent measurements to find x

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Applications - State Estimation

• Objective estimate V1 and V2
• In practice our measurement is corrupted by noise
  n z ztrue n
• Find an estimate x for the true value of state
  xtrue where ztrue A xtrue
• Error e z-Ax
• Find x to minimize e2 eTe (z-Ax)T(z-Ax)

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Applications - State Estimation

• Objective estimate V1 and V2
• In practice our measurement is corrupted by noise
  n z ztrue n
• Find an estimate x for the true value of state
  xtrue where ztrue A xtrue
• Error e z-Ax
• Find x to minimize e2 eTe (z-Ax)T(z-Ax)

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Applications - State Estimation

• Find x to minimize u(x) e2 eTe
  (z-Ax)T(z-Ax)
• ?u 0 -2 AT z 2ATAx
• ATA z AT z x (ATA)-1ATz when
  inverse exists

Often we weight the measurement to reflect meter
accuracy Find x to minimize u(x)
(z-Ax)T W (z-Ax) ATWA z AT W z x
(ATA)-1AT W z when inverse exists
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Applications - State Estimation
4.27 R1R3R51.5 ohm -1.71 R2R4 1
ohm z 3.47 2.5 Measurement see text or z3
G1G2G3 -G3 -1 V1G1 z1 (V1-z3)G1
z4 -G3 G3G4G5 V2G5 z2
(V2-z4)G5
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Applications - State Estimation
A
4.27 -1.71 z 3.47 2.5
x (ATA)-1ATz
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Applications - State Estimation
A
4.27 -1.71 z 3.47 2.5
x (ATA)-1ATz
4.245 -1.73 z 3.526 2.449
e
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Applications - State Estimation
A
4.27 1.71 z 3.47 2.5
V1 10.54 V20.122
x (ATA)-1ATz
4.834 0.569 z 3.29 0.975
Systematic statistical tests Allow bad data
detection
-0.564 1.141 e 0.18 1.525
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Applications - State Estimation
A
4.27 1.71 z 3.47 2.5
V19.881 V25.135
1 0 W 1 1
x (ATWA)-1ATWz
Removing bad data can recover the estimate
4.234 -1.77 z 3.53 2.476
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Applications - State Estimation
In many power system applications the measurement
model is non linear since we measure power and
reactive power z h(x) X is voltage magnitudes
and phase angles Ue2 (z-h(x))T W
(z-h(x)) WR-1 ?U -HT W (z-g(x)0 where Hx
Jxh is the Jacobian Use NR to solve this
nonlinear equation
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Applications - State Estimation
State x V3 ?2 ?3
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Applications - State Estimation
State x V3 ?2 ?3
H
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Applications - State Estimation
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Applications - State Estimation
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Summary Optimization Methods Formulation
and Application Economic Dispatch OPF SCOPF
State Estimation