Optimal Power Flow
Junjian Qi
Dept. of Electrical Engineering & Computer Science
University of Tennessee, Knoxville
Nov. 25, 2013
Economic Dispatch
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Objective: minimize the cost of generation
Constraints
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Equality constraint: load generation balance
Inequality constraints: upper and lower limits on
generating units output
May result in unacceptable flows or voltages in the
network
Optimal Power Flow (OPF)
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Optimization problem
Classical objective function
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Equality constraints
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Minimize the cost of generation
Power flow equations, power balance at each node
Inequality constraints
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Network operating limit (line flows, voltages)
Limit on control variables
Optimal Power Flow (OPF)
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Parameters
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Known characteristics of the system
Assumed constant
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Network topology
Network parameters (R, X, B, flow and voltage limits)
Generator cost functions
Generator limits
Vector of parameters: y
Optimal Power Flow (OPF)
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Decision variables (control variables)
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Active power output of the generating units
Voltage at the generating units
Position of the transformer taps
Status of the switched capacitors and reactors
Control of power electronics (HVDC, FACTS)
Amount of load disconnected
Vector of control variables: u
Optimal Power Flow (OPF)
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State variables
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Describe the response of the system to changes in the
control variables
Magnitude of voltage at each bus
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Except generator busses, which are control variables
Angle of voltage at each bus
Vector of state variables: x
Optimal Power Flow (OPF)
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Classical objective function
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Minimize total generating cost: min
u
N
PG
Ci (PGi )
i=1
Many other objective functions are possible
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N
PG
Minimize active power loss:
PGi −
i=1
N
PG
Minimize reactive power loss
i=1
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N
PD
PDi
i=1
N
PD
QGi −
i=1
Minimize number of controls rescheduled
Miminize load shedding
QDi
Optimal Power Flow (OPF)
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Equality constraints:
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Power balance at each node–power flow equations
For AC Power Flow:
PGk − PDk =
QGk − QDk =
N
X
i=1
N
X
Vk Vi [Gki cos(θk − θi ) + Bij sin(θk − θi )] (1)
Vk Vi [Gki sin(θk − θi ) − Bij cos(θk − θi )] (2)
i=1
For DC Power Flow:
F =Bθ
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Compact expression: G(x , u, y ) = 0
(3)
Optimal Power Flow (OPF)
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Inequality constraints:
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Limits on the control variables: umin ≤ u ≤ umax
Operating limits on flows: |Fij | ≤ Fijmax
Operating limits on voltages: Vmin ≤ V ≤ Vmax
Compact expression: H(x , u, y ) ≤ 0
Optimal Power Flow (OPF)
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min
f (u)
u
(4)
s.t. G(x , u, y ) = 0
H(x , u, y ) ≤ 0
(5)
(6)
Linear programming
Nonlinear programming
Mixed integer linear programming
Mixed integer nonlinear programming
Security-Constrained OPF
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Conventional OPF only guarantees that the operating
constraints are satisfied under normal operating
conditions
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–All lines in service
This does not guarantee security
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–Must consider N-1 contingencies
Assume lime limits for Line 1-3, 1-2, and 2-3 are 300 MW, 100
MW, and 220 MW
Base case OPF solution
Contingency case
Security-Constrained OPF
min
f (u0 )
u
(7)
s.t. G(x0 , u0 , y0 ) = 0
H(x0 , u0 , y0 ) ≤ 0
(8)
(9)
0
(
∀k
G(xk , u0 , yk ) = 0
H(xk , u0 , yk ) ≤ 0
Subscript 0 indicates value of variables in the base case
Subscript k indicates value of variables for contingency k
Stability-Constrained OPF
Reduced network:
Swing equations:
δ˙i = ωR (ωi − 1)
1
(Pmi − Pei )
ω̇i =
Mi
where Pei = Ei
X
j
(10)
(11)
Ej [Bij sin(δi − δj ) + Gij cos(δi − δj )] (12)
Stability-Constrained OPF
Initial value:
!∗
Ei ∠δi0 − Vi ∠θi
S̃ = Ṽi Ĩ = Vi ∠θi
jxd0
Ei Vi sin(δi0 − θi )
Ei Vi cos(δi0 − θi ) − Vi2
=
+
j
xd0
xd0
Equality constraints for initial value:
∗
Ei Vi sin(δi0 − θi )
− PGi = 0 ∀i ∈ G
xd0
Ei Vi cos(δi0 − θi ) − Vi2
− QGi = 0 ∀i ∈ G
xd0
ωi0 = 1 ∀i ∈ G
(13)
(14)
(15)
(16)
Stability-Constrained OPF
Discretize swing equations with trapezoidal rule:
δit+1 −
ωit+1 − ωit −
where Peit = Ei
∆t
ωR (ωit+1 + ωit − 2) = 0
2
∀t ∈ T , ∀i ∈ G
∆t 1
(Pmi − Peit+1 + Pmi − Peit ) = 0
2 Mi
∀t ∈ T , ∀i ∈ G
X
(17)
(18)
Ej [Bijt sin(δit − δjt ) + Gijt cos(δit − δjt )] (19)
j
Gijt and Bijt are different for during-fault and post-fault states
Stability-Constrained OPF
For each time step t, calculate δ t of the OMIB system
δt =
1 X
1 X
Mi δit −
Mi δit
MC i∈GC
MNC i∈GNC
(20)
Inequality constraint for transient stability
δ t ≤ δ max ∀t ∈ T
(21)
Stability-Constrained OPF
min
X
PG
2
ai + bi PGi + ci PGi
(22)
i∈G
s.t.
PG − PL − P(V , θ) = 0
(23)
QG − QL − Q(V , θ) = 0
(24)
max
(25)
max
(26)
max
Pg
(27)
max
(28)
F (V ,θ) ≤ F
V
min
min
PG
min
QG
≤V ≤ V
≤PG ≤
≤QG ≤ Qg
Ei Vi sin(δi0 − θi )
− PGi = 0 ∀i ∈ G
x0
(29)
d
Ei Vi cos(δi0 − θi ) − Vi2
x0
− QGi = 0 ∀i ∈ G
(30)
d
0
ωi = 1 ∀i ∈ G
t+1
δi
t+1
ωi
−
t
ωi
−
−
∆t
t+1
ωR (ωi
2
∆t 1
2 Mi
(Pmi −
t
+
t
ωi
t+1
Pei
δ ≤δ
max
(31)
− 2) = 0 ∀t ∈ T , ∀i ∈ G
+ Pmi −
∀t ∈ T
t
Pei )
= 0 ∀t ∈ T , ∀i ∈ G
(32)
(33)
(34)
Stability-Constrained OPF
Conventional OPF
Stability-constrained OPF
Algorithms
Challenges:
I Nonlinear
I Nonconvex
Algorithms:
I Quadratic programming
I Lagrange relaxation
I Interior point method
I Artificial neural network
I Tabu search algorithm
I Genetic algorithm
I Particle swarm
Many are based on Karush-Kuhn-Tucker (KKT) necessary
conditions and can only guarantee a locally optimal solution
Convexifying OPF
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Solve a relaxed dual problem, which is convex
programming
Derive a necessary and sufficient condition to guarantee
zero duality gap
Source: Daniel Molzahn
Further Reading
Stability-Constrained OPF:
I D. Gan, R. Thomas, and R. Zimmerman.
Stability-constrained optimal power flow. IEEE
Transactions on Power Systems, vol. 15, May 2000.
I R. Zárate-Miãno, T. V. Custem, F. Milano, and A. J.
Conejo. Securing transient stability using time-domain
simulations within an optimal power flow. IEEE
Transactions on Power Systems, vol. 25, Feb. 2010.
I D. Ernst, D. Ruiz-Vega, M. Pavella, P. Hirsch, and D.
Sobajic. A unified approach to transient stability
contingency filtering, ranking and assessment. IEEE
Transactions on Power Systems, vol. 16, Aug. 2001.
Convexifying OPF:
I J. Lavaei and S. Low, Zero duality gap in optimal power
flow problem. IEEE Transactions on Power Systems, vol.
27, Feb. 2012.