This presentation describes contingency analysis due to generator and line outages using distribution factors. Also, ranking of contingencies is discussed.

1. Power System Security
Power System Operation & Control
Course Code: EEE 4243
Md. Mahmudul Hasan
Lecturer, Dept. of Electrical & Electronic Engineering
Rajshahi University of Engineering & Technology
E-mail: mh.mahmud.ruet@gmail.com
mahmud@eee.ruet.ac.bd
Cell: 01837-540698

2. Reference:
Antonio Gomez-Exposito,
"Electric energy systems:
analysis and operation." CRC
Press, 2008. Chapter- 6.

3. Ability of a power system to maintain the desired state without violating any
of the imposed operational limits (e.g. bus voltages and line flows) against
predictable changes (demand and generation evolution) and unpredictable
events (called contingencies) likely to occur during real‐time operation.
• Operating Constrains: Operational limits on
system variables and apparatus, for instance voltage
limit, generator loading limit, transmission line
thermal limit, tap position limit and so on.
• Load Constraints: Mainly refer to customers’ total
power demand.
• Security Constraints: Minimum reserve margin in
generation and transmission.
Power
System
Operating
Constrains
Security
Constrains
Load
Constrains
Power System Security
Security Related Constrains

4. Power System Security Related States

5. Contingency can be analyzed:
• Using Distribution Factors: Can only identify overloads i.e. real power limit violations in transmission
lines by an approximate method.
• A fast decoupled load flow analysis: Only for the contingencies, selected through a screening process.
This method can identify real power flow as well as voltage violations.
Contingency Analysis Techniques
Any unpredictable event in power system is known as Contingency.
Ranking of Contingencies:
• Ranking the contingencies in descending order through application of distribution factors or
• Identifying the severity potential of any contingency firstly running a fast decoupled load flow only for
one or two iterations in the post contingency state.

6. Computation of Line Power Flow

7. …(4)
…(5)
…(6)
From eqns (4), (5) and (6),
Using eqns (7)and (8),
Taking complex conjugate,
…(9)
…(10)
From eqns (9) and (10),
= 𝑉(cos 𝜃 + 𝑗 sin 𝜃) leading to,
This is the conjugate of complex line power flow.
…(7)
…(8)

8. Lets recall the eqn. below,
Separating the real and imaginary parts,
Real power flow from
bus- i to bus- k
Reactive power flow
from bus- i to bus- k
Similarly,
Real power flow from
bus- k to bus- i
Reactive power flow
from bus- k to bus- i

9. DC Load Flow

10. Lets recall the eqn. below:
We know, in matrix form. Hence,
Real power flow from
bus- i to bus- k
𝑌𝑖𝑗 cos 𝜃𝑖𝑗 = 𝐺𝑖𝑗 and 𝑌𝑖𝑗 sin 𝜃𝑖𝑗 = 𝐵𝑖𝑗
Now, real power flow from bus-i to bus-j:
𝑃𝑖𝑗 = − 𝑉𝑖
2
𝑌𝑖𝑗 cos 𝜃𝑖𝑗 + 𝑉𝑖 𝑉
𝑗 𝑌𝑖𝑗 cos 𝜃𝑖𝑗 − 𝛿𝑖 − 𝛿𝑗
= − 𝑉𝑖
2
𝐺𝑖𝑗 + 𝑉𝑖 𝑉
𝑗 𝑌𝑖𝑗 cos 𝜃𝑖𝑗 cos 𝛿𝑖 − 𝛿𝑗 + sin 𝜃𝑖𝑗 𝑠𝑖𝑛 𝛿𝑖 − 𝛿𝑗
= − 𝑉𝑖
2
𝐺𝑖𝑗 + 𝑉𝑖 𝑉
𝑗 𝑌𝑖𝑗 cos 𝜃𝑖𝑗 cos 𝛿𝑖 − 𝛿𝑗 + 𝑌𝑖𝑗 sin 𝜃𝑖𝑗 𝑠𝑖𝑛 𝛿𝑖 − 𝛿𝑗
= − 𝑉𝑖
2
𝐺𝑖𝑗 + 𝑉𝑖 𝑉
𝑗 𝐺𝑖𝑗cos 𝛿𝑖 − 𝛿𝑗 + 𝐵𝑖𝑗 𝑠𝑖𝑛 𝛿𝑖 − 𝛿𝑗
= 𝑉𝑖 𝑉
𝑗 𝐺𝑖𝑗cos𝛿𝑖𝑗 +𝐵𝑖𝑗 sin 𝛿𝑖𝑗 − 𝑉𝑖
2
𝐺𝑖𝑗
Lets use θ instead of δ for voltage phase angle for your convenience to study form “Electric Power Systems” by
Antonio Gomez.
∴ 𝑷𝒊𝒋 = 𝑽𝒊 𝑽𝒋 𝑮𝒊𝒋𝐜𝐨𝐬𝜽𝒊𝒋 +𝑩𝒊𝒋 𝐬𝐢𝐧 𝜽𝒊𝒋 − 𝑽𝒊
𝟐
𝑮𝒊𝒋

11. DC Load Flow
 A linear approximation between P and 𝜃.
 𝑉𝑖 = 1 at all the buses.
 Losing the capability to track reactive power and voltage related data.
We know, real power flow from bus-i to bus-j:
𝑃𝑖𝑗 = 𝑉𝑖 𝑉
𝑗 𝐺𝑖𝑗cos𝜃𝑖𝑗 +𝐵𝑖𝑗 sin 𝜃𝑖𝑗 − 𝑉𝑖
2
𝐺𝑖𝑗
Put 𝑉𝑖 = 1 and 𝑉
𝑗 = 1,
Now, element wise, 𝐺 + 𝑗𝐵 =
−𝟏
𝒓+𝒋𝒙
=
− 𝒓−𝒋𝒙
𝒓+𝒋𝒙 𝒓−𝒋𝒙
=
− 𝒓−𝒋𝒙
𝒓𝟐+𝒙𝟐 =
−𝒓
𝒓𝟐+𝒙𝟐 + 𝒋
𝒙
𝒓𝟐+𝒙𝟐
∴
In transmission networks values of r/x < 3. The error arising when 𝐵𝑖𝑗 is replaced by 1/𝑥𝑖𝑗 is lower than 1%,
yielding:
Conventional Load Flow

12. Writing the DC power flow equation,
DC Load Flow
in matrix form:
Diagonal matrix of
branch reactance
Transpose of
branch-to-node
incidence matrix
L1 L2 L3 L4 L5 L6
Bus power injections,
1
2
3
4
Branches
Nodes
Bus Power Injections
𝑃1 = 𝑃𝐿1 + 𝑃𝐿2 + 𝑃𝐿3 𝑃3 = −𝑃𝐿2 + 𝑃𝐿5
𝑃2 = −𝑃𝐿1 + 𝑃𝐿4 𝑃4 = −𝑃𝐿3 − 𝑃𝐿5 + 𝑃𝐿6
Corresponds to that of a resistive-like DC circuit composed of
reactance, in which active powers play the role of DC current
injections and phase angles that of DC voltages (I=BV, B=1/X),
which explains its name – DC Load Flow .
Bus-5
is Slack

13. Contingency Analysis Based on
Distribution Factors
Security Assessment:
Contingency Analysis

14. Line Flow Change Due to Generator Outage
Now, we know, the active power injection matrix, P=B𝜃 → 𝜃 = 𝐵−1
𝑃. Also,
** The above expression is useful to quickly analyze power flow changes arising from branch or generator outages.
∴ (𝑆𝑓 is the matrix of sensitivities between branch power flows and injected powers)
 Using Superposition principle:
 Injection Distribution Factor:
 The flow increase in a branch, ∆𝑷𝒎𝒏 due to generator outage:
• If the lost generation is assumed by the slack bus,
• If the outage generation is shared among the remaining generators to model the response of the automatic
generation control (AGC), “sharing” factors must be used

15. Example -1
Assume the outage of generator at bus- 3. Find the change in line
power flows. Consider bus- 5 as slack.
Solution:
Bus admittance matrix
neglecting conductance
𝑆𝑚𝑛,3 i.e., distribution factor between each line and bus- 3

16. If the slack bus fully compensates for the outaged generation,
Example -1 (Continued)
Hence, the increase in line power flow,
∴ ∆𝑃𝑓 = 𝑆𝑓∆𝑃𝑖=
0
0
−1000
0
=
−414
483
−69
−414
−517
−586
** In practice, in case of a generator outage, several generators assume the power imbalance due to the action of
the Automatic Generation Control.
Then, let us assume that the power imbalance is shared by the remaining generators in proportion to PG max.
Thus, 600 MW will go to generator 4 (PG max 4 = 1500 MW) and 400 MW to generator 5 (PG max 5 = 1000 MW).
This yields, . This is postcontingency power flow using sharing factors
A𝑛𝑑 𝐶𝑜𝑛𝑠𝑒𝑞𝑢𝑒𝑛𝑡𝑙𝑦, ∆𝑃𝑓 = 𝑆𝑓∆𝑃𝑖=
0
0
−1000
600
=
−207
441
−234
−207
−559
−193

17. Flow Change Due to Line Outage
Distribution factors due to line outage can
be obtained by modeling the outage of the
branch using two fictitious injections at
both ends.
Fig. Modeling a branch outage using fictitious injections.
** The fictitious injections must coincide with the power flow after the outage:
Then, the flow through branch mn after branch ij outage is obtained as
∴ Corresponding distribution factor:
∆𝑃𝑖𝑗
𝑚𝑛 = 𝑆𝑚𝑛,𝑖 − 𝑆𝑚𝑛,𝑗 ∆𝑃𝑖 = 𝑆𝑚𝑛,𝑖 − 𝑆𝑚𝑛,𝑗

18. Example -2
Assume the outage of line 1-3. Find the change in line power flows.
Consider bus- 5 as slack.
Solution:
Bus admittance matrix
neglecting conductance

19. Distribution factors can be obtained using two fictitious power injections at nodes 1 and 3 with
Example -2 (Continued)
Now, we have the matrix of sensitivities between branch power flows and injected powers, 𝑆𝑓
𝐿𝑖𝑛𝑒 𝑓𝑙𝑜𝑤 𝑐ℎ𝑎𝑛𝑔𝑒 𝑐𝑎𝑛 𝑏𝑒 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑑𝑖𝑟𝑒𝑐𝑡𝑙𝑦, ∆𝑃𝑖𝑗
𝑚𝑛 = 𝑆𝑚𝑛,𝑖 − 𝑆𝑚𝑛,𝑗 ∆𝑃𝑖 = 𝑆𝑚𝑛,𝑖 − 𝑆𝑚𝑛,𝑗
Then, the distribution factors are obtained from the original sensitivity matrix, Sf, as,
𝑺𝑳𝟐,𝟏 𝑺𝑳𝟐,𝟐 𝑺𝑳𝟐,𝟑 𝑺𝑳𝟐,𝟒
Useless

20. Ranking of Contingencies

21. A commonly used method to detect critical contingencies is based on determining a contingency ranking in
descendent order of severity. A ranking index is used to quantify the loading level of the system after a given
outage.
Ranking Index
where Pfk is the branch flow of element k, from a total of b lines and transformers,
obtained approximately using distribution factors. In this case, the ranking index is
just the average rate of lines and transformers.
 In practice, several versions of ranking indexes have been proposed, including weighting factors to give more
importance to relevant elements.
 Once the ranking indexes have been obtained for all possible contingencies, they are classified in descendent
order. In this way, the analysis begins with the a priori most critical contingency, going down in the list until a
non-problematic contingency is analyzed .
 The main drawbacks of methods based on ranking indexes are
o Errors inherent to distribution factors
o Possibility of “masking” a problematic contingency as a result of “condensing” the loading rates of all
branch elements in just one value, that is, the ranking approach might give priority to a contingency
that results in “slight” overloads on several elements instead of a critical contingency in terms of
overloading magnitude.

22. Contingency Analysis Based on Ranking Index
Let us consider all outages except the slack generator, as
it is responsible for supplying any generation imbalance.
Then, obtaining the postcontingency flows by means of
the corresponding distribution factors for each one of
the contingencies, the following ranking indexes are
obtained:
In consequence, postcontingency states can be
classified in descendent RI order: L2, G3, L3, L4,
G4, L1, L6, and L5.

23. A detailed analysis of postcontingency states using a load flow provides
the following results:
Contingency Analysis Based on Fast Decoupled Load Flow
The ranking analysis would have
stopped after detecting a non
overloaded case (G3), leaving without
attention some critical contingencies:
L3 and L4 outages. This problem is
usually known as the masking effect
and is inherent to ranking methods.
 Distribution factors totally ignore voltage problem (assume those 1
pu). So alternative screening method is required which could be a
single iteration of FDLF on the network considering N‐1 condition.
 Obviously, the main advantage of the screening techniques over
ranking index based techniques is the fact that all contingencies
are analyzed, though approximately, without making any previous
selection.