Module 5
Contingency Analysis
5.1
Introduction and concept of linear sensitivity factors
In Chapter 2, we have discussed about the methods for analyzing the power system performance at a
particular operating point. However, for practical system operation, apart from ensuring the satisfactory operation of the system at a particular operating condition, it is also equally important to make
sure that the system operates with adequate level of security. Broadly, the term ‘security’ implies
the ability of the system to operate within system constraints (on bus voltage magnitudes, current
and power flow over the lines) in the event of outage (contingency) of any component (generator or
transmission line). Now, if the system is operating at high loading (light loading) conditions, then
the post-contingency system condition would be highly stressed (lightly stressed). Therefore, the
post contingency values of different quantities (voltages, current/power flow) depend on the present
operating condition. In case the post-outage (post contingency) does not involve any violation of
any operating constraints, the system is said to be operating securely. Otherwise, the system is said
to enter an emergency operating condition. Therefore, for detecting the possibility of appearance
of emergency operating conditions, analysis of the post-contingency scenario (henceforth termed as
contingency analysis) of the system needs to be carried out.
Now, in any practical sized power system, there is a very large number of elements. Hence, for
carrying out contingency analysis, outages of all these elements (preferably) need to be carried out
one-by-one corresponding to any particular operating condition. However, in any power system, the
operating point of the system changes quite frequently with change is loading/generating conditions.
With the change in system operating conditions, the contingency analysis exercise needs to be
carried out again at the new operating point. Thus, for proper monitoring of system security, a
large number of outage cases need to be simulated repeatedly over a short span of time. Ideally,
these outage cases should be studied with the help of full AC load flow solutions. However, analysis
of thousands of outage cases with full AC power flow technique will involve a significant amount of
computation time and as a result, it might not be possible to complete this entire exercise before the
new operating condition emerges. Therefore, instead of using full non-linear AC power flow analysis,
approximate, but much faster techniques based on linear sensitivity factors are used to estimate the
post contingency values of different quantities of interest. The basic concept of sensitivity factors
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is described below. Essestially, the linear sensitivity factors approximately estimate the changes
in different line flows for any particular outage condition without the need of full AC power flow
solution.
Basically, there are two types of sensitivity factors and these are:
a. Generation outage sensitivity factor (GOSF)
b. Line outage sensitivity factor (LOSF)
GOSF relates the approximate change in power flow in line ‘i-j’ (i.e. between bus ‘i’ and ‘j’) due
to the outage of generator at bus ‘k’, whereas LOSF helps to calculate the approximate change in
power flow in line ‘i-j’ due to outage of line ‘m-n’.
The generation outage sensitivity factor is defined by,
αijk =
∆fij
∆Pk
(5.1)
k
where, αij
→ GOSF of line ‘i-j’ for generation change at bus ‘k’
∆fij → Change in power flow in line ‘i-j’
∆Pk → Change in generation at bus ‘k’
k
The factor αij
denotes the sensitivity of the line flow on line ‘i-j’ due to change in generation
at bus ‘k’. In equation (5.1), it is assumed that the generation lost at bus ‘k’ would be exactly
compensated by the reference or slack bus. Now, if the generation at bus ‘k’ was generating an
amount of power equal to Pk0 , then to represent the outage condition, ∆Pk = −Pk0 .
Hence, the new power flow over the line ‘i-j’ would be given as,
fijn = fij0 + ∆fij = fij0 + αijk ∆Pk = fij0 − αijk Pk0
(5.2)
k
The factor αij
would be pre-calculated and stored in the memory. As we will see later, the values
k
of αij depend only on the network parameters and therefore, are constant. However, it should be
k
m
noted that for any particular line ‘i-j’, the factors αij
and αij
(for generation outage at bus ‘m’) are
different and therefore need to be pre-calculated separately. Once these factors are pre-calculated
and stored, the new values of line flow over any line can easily be estimated very quickly from
equation (5.2). If the new power flow over any line is found to be more than the corresponding limit,
then the operator can be alerted for taking an appropriate pre-emptive action.
In equation (5.2), it is assumed that the lost generation at bus ’k’ would be taken up by the
slack bus. However, it is also quite possible that the lost generation would be compensated by
all the remaining ‘on-line’ generators combinedly, in which, each of the ‘on-line’ generators would
take up some fraction of the lost generation in some particular ratio. One of the most frequently
used methods assumes that the ‘on-line’ generators share the lost generation in proportion to their
maximum MW rating. Thus, the proportion of generation picked up by generation ‘g’ is given by
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g ≠ k,
γgk =
Pgmax
(5.3)
M
∑ Pamax
a=1
≠k
Where, M → Total number of generators in the system
γgk → Proportionality factor for generation ‘g’ to pick up generation when unit ‘k’ fails
Pamax → Maximum MW rating for generator ‘a’.
Now, as the sensitivity factors shown in equation (5.1) are linear in nature, the effects of simultaneous generation change in several generators on a particular line can be obtained by following
superposition principle. Hence, the new line flow in the line ‘i-j’ becomes,
m
fij(n) = fij(0) + αijk ∆Pk − ∑ αija ∆Pa γak
(5.4)
a=1
In equation (5.4) it is assumed that no remaining ‘on-line’ generation hits the generation limit.
The line outage distribution factors are also defined similarly. The LOSF is defined by,
βij, mn =
∆fij
(5.5)
(0)
fmn
Where, βij, mn →Line outage distribution factor for line ‘i-j’ under outage of line ‘m-n’.
(0)
fmn
→ Power flow over line ‘m-n’ in the pre-outage condition.
Therefore, for the outage of line ‘m-n’, the new flow over line ‘i-j’ is given by,
(0)
fij(n) = fij(0) + βij, mn fmn
(5.6)
Again, as we will show later, the factors βij,mn are constant as they are dependent only on the line
parameters. Therefore, they would be pre-calculated and stored in the memory. As a result, for the
outage of any line ‘m-n’, the new power flows over all the other lines can be estimated very quickly.
5.2
DC load flow and generation outage distribution factor
We have already discussed the concepts of linear sensitivity factors. These factors are calculated
based on the concept of DC power flow and hence, let us first have a look at DC power flow
technique.
5.2.1
DC power flow
From FDLF method, we know, ∆P = [B ] ∆θ , where each elements of the matrix [B ] is negative
of the imaginary parts of the corresponding elements of the ȲBUS bus matrix. Now, in DC power
′
′
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flow analysis, apart from using the above decoupled relation between ∆P and ∆θ , several other
simplifying assumptions are also taken as follows:
a. The system is lossless (i.e. line resistance is neglected) and therefore, each line is represented
by its reactance only.
b. The voltage of each bus is maintained at the rated voltage, i.e. 1.0 p.u.
c. For any line ‘m-n’, the angular difference between its terminal buses is quite small, and hence,
cos θm ≈ cos θn (as θm ≈ θn ) and sin(θm − θn ) ≈ (θm − θn ) rad (as θm − θn ≈ 0) .
With these assumptions, the power flow over a line becomes,
Pij =
Vi Vj
1
sin(θi − θj ) ≈
(θi − θj ) (p.u)
xij
xij
(5.7)
In equation (5.7), the quantity xij denotes the reactance of the line ‘i-j’. From this equation it
is observed that the line power flow is basically a linear combination of the terminal bus voltage
angles. Moreover, the current flow over the line ‘i-j’ is given by,
V̄i − V̄j
I¯ij =
jxij
(Vi cos θi − Vj cos θj ) + j(Vi sin θi − Vj sin θj )
=
jxij
1
1
=
(sin θi − sin θj ) ≈
(θi − θj )
xij
xij
(5.8)
Equation (5.8) has been written under an added assumption that both the angles θi and θj are
individually quite small in magnitude. From equations (5.7) and (5.8) it is observed that in DC
power flow model, the expressions of line power flow and line current are same in per unit. We will
utilize this fact for computing LOSF in future.
In the next lecture, we will discuss a method for calculating GOSF.
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