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Smooth Formulation of Hard Constraints in Voltage Stability Analysis Adriano Lima Abrantes August 27, 2016 1
Smooth Formulation of Hard Constraints in Voltage Stability AnalysisAdriano Lima AbrantesAugust 27, 2016
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ScopeStatic Analysis of Voltage Stability
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MotivationMaximum Loading Point
● Saddle-Node Bifurcation
● Limit Induced Bifurcation
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Objectives● Provide a set of smooth equations for the Power Flow problem with
hard constraints
● Apply the point-of-collapse method to compute the maximum
loading point regardless of the bifurcation type
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Electric Power System Model
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PV Curve
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Point-of-collapse Method
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Voltage Regulation and Reactive Power Limits
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Voltage Regulation and Reactive Power Limits
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Constraint Exchange Point
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● Increases state sensitivity in
respect to load
● There still is some Load
Margin left
Limit Induced Bifurcation
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● There is no Load Margin left
● Grid is unable to attend any
additional load
Smooth Formulation of Hard Constraints
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Smooth Formulation of Hard Constraints
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3 Bus System
Bus V θ Pinj Qinj
1 1 0
2 1 0
3 -0.8λ -0.6λ
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3 Bus System: Saddle-Node Bifurcation Case
V (p
u)
θ (r
ad)
Qge
n (p
u)
λ (pu)
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3 Bus System: Limit Induced Bifurcation Case
V (p
u)
θ (r
ad)
Qge
n (p
u)
λ (pu)
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Eigenvalue Analysis
λ (loading)
Rea
l par
t of
eig
enva
lue
of D
xf
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45 Bus SystemA reduced system that models the Southern part of the Brazilian National Interconnected System
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45 Bus System: Saddle-Node Bifurcation Case
V (p
u)
θ (r
ad)
Qge
n (p
u)
λ (pu)
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45 Bus System: Limit Induced Bifurcation Case
V (p
u)
θ (r
ad)
Qge
n (p
u)
λ (pu)
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Summary● The smooth formulation provides a differentiable set of equations
for the power flow problem with hard constraints
● This allows the application of the point-of-collapse method for
computing the maximum loading point even when limit induced
bifurcation occurs
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