Sensitivity based Thevenin index for voltage stability assessment considering N-1 contingency

1. A method for determining long-term voltage stability in a power system, comprising:

estimating post-contingency operating point(s) of the power system considering PV-PQ transitions based on the current operating point;

predicting a sensitivity based Thevenin index (STI) for each contingency using the estimated operating condition; and

determining a voltage stability assessment (VSA) considering N−1 contingency, wherein the current operating states {tilde over (θ)},{tilde over (V)}, further comprising determining post-contingency operating states as:

[

θ

c

V

c

]

=

[

θ

~

V

~

]

⁡

[

f

θ

f

V

g

θ

g

V

]

-

1

⁢

Δ

⁢

⁢

F

inj

where ({tilde over (θ)}, {tilde over (V)}) is an operating point under base case; (θ c ,V c ) is an operating point following a contingency; Δ F inj is a vector containing the active and reactive bus injection changes which are numerically equal to the active and reactive line flow on an outage branch; f θ and f V represents the partial derivative of active power injection expression with respect to voltage angle θ and voltage magnitude V; g θ and g V denotes the partial derivative of reactive power injection expression with respect to voltage angle θ and voltage magnitude V; and

operating a power generator adapted to be connected to a power grid and controlled by a processor.

2. The method of claim 1 , further comprising:

obtaining current operating states;

estimating a post-contingency operating point θ c ,V c for every considered contingency, where θ c and V c represent the predicted bus voltage angle and bus voltage magnitude after a contingency;

determining corresponding sensitivities based on the presence of PV-PQ transition following the contingency; and

determining the STI for each considered contingency.

3. The method of claim 1 , comprising estimating an operating point after a foreseen transmission contingency.

4. The method of claim 1 , comprising construction of a Jacobian matrix based on a system topology with a contingency line ij in an outage.

5. The method of claim 1 , comprising determining corresponding sensitivities as one predictor step of continuation of power flow (CPF) when dλ is a continuation parameter with:

[

f

θ

f

V

f

λ

P

g

θ

g

V

g

λ

Q

0

0

1

]

⁡

[

d

⁢

⁢

θ

dV

d

⁢

⁢

λ

]

=

[

0

0

1

]

where dλ represents the load increment; dθ and dV denote the change of voltage angle and voltage magnitude respectively; f p λ ,g p λ indicate the amount of active and reactive power injection change as a function of dλ.

6. The method of claim 5 , comprising obtaining sensitivities when there are generators reaching their reactive power limit after the contingency as:

[

f

θ

f

V

f

V

i

f

λ

P

g

θ

g

V

g

V

i

g

λ

Q

g

θ

i

g

V

i

g

V

i

i

g

λ

Q

,

i

0

0

0

1

]

⁡

[

d

⁢

⁢

θ

dV

dV

i

d

⁢

⁢

λ

]

=

[

0

0

0

1

]

where bus i is the bus which transforms from PV bus to PQ bus; f V i is the partial derivative of active power injection expression at other PV and PQ buses with respect to bus voltage magnitude at bus i, i.e., V i ; g V i is the partial derivative of reactive power injection expression at other PQ buses with respect to bus V i ; g θ i , g V i , g V i i , represent the partial derivative of reactive power injection expressions at bus i with respect to θ,V and V i ; g λ Q,i is the partial derivative of reactive power injection expression at bus i with respect to the load scaling factor A.

7. The method of claim 1 , comprising determining STI for each contingency and load bus in parallel.

8. The method of claim 1 , comprising determining a plurality of contingencies in parallel.

9. A system to control a power system, comprising:

a power generator;

a power grid coupled to the power generator; and

a processor to control the power grid with code for:

estimating post-contingency operating point(s) of the power system considering PV-PQ transitions based on the current operating point;

predicting a sensitivity based Thevenin index (STI) for each contingency using the estimated operating condition;

determining a voltage stability assessment (VSA) considering N−1 contingency; and

determining corresponding sensitivities as one predictor when dλ is a continuation parameter with:

[

f

θ

f

V

f

λ

P

g

θ

g

V

g

λ

Q

0

0

1

]

⁡

[

d

⁢

⁢

θ

dV

d

⁢

⁢

λ

]

=

[

0

0

1

]

where dλ represents the load increment, dθ and dV denote the change of voltage angle and voltage magnitude respectively, f p λ ,g p λ indicate the amount of active and reactive power injection changes as a function of dλ; and where f θ and f V represents the partial derivative of active power injection expression with respect to voltage angle θ and voltage magnitude V−g θ and g V denotes the partial derivative of reactive power injection expression with respect to voltage angle θ and voltage magnitude V.

10. The system of claim 4 , further comprising code for:

obtaining current operating states;

estimating a post-contingency operating point θ c ,V c considering the presence of PV-PQ transition for every considered contingency;

determining corresponding sensitivities following the contingency; and

determining the STI for each considered contingency.

11. The system of claim 9 , comprising code for obtaining sensitivities when there are PV-PQ transitions as:

[

f

θ

f

V

f

V

i

f

λ

P

g

θ

g

V

g

V

i

g

λ

Q

g

θ

i

g

V

i

g

V

i

i

g

λ

Q

,

i

0

0

0

1

]

⁡

[

d

⁢

⁢

θ

dV

dV

i

d

⁢

⁢

λ

]

=

[

0

0

0

1

]

where bus i is the bus which transforms from PV bus to PQ bus; f V i is the partial derivative of active power injection expression at other PV and PQ buses with respect to bus voltage magnitude at bus i, i.e., V i ; g V i is the partial derivative of reactive power injection expression at other PQ buses with respect to bus V i ; g θ i ,g V i ,g V i i represent the partial derivative of reactive power injection expressions at bus i with respect to θ,V and V i ; g λ Q,i is the partial derivative of reactive power injection expression at bus i with respect to the load scaling factor λ.

12. The system of claim 9 , comprising code for determining STI for each contingency and load bus in parallel.

13. The system of claim 9 , comprising code for determining a plurality of contingencies in parallel.