5.5.1

Example of contingency analysis for two line outages

As an example, let us consider outage of two lines in the IEEE-14 bus system. Towards this goal, let us assume that line no. 3 (let it be considered as line ‘e-f’) and 17 (let it be considered as line ‘g-h’) have been taken out of the system. Subsequently, for any line ‘i-j’ (in the set of lines which are still remaining in the system), the factors LOSFM1 and LOSFM2 are calculated as follows.

LOSF M 1 =

Lij,ef + Lij,gh Lgh,ef 1 − Lef,gh Lgh,ef

(5.63)

LOSF M 2 =

Lij,gh + Lij,ef Lef,gh 1 − Lef,gh Lgh,ef

(5.64)

Now, under the assumption of DC power flow, for contingency calculation, the impedances are replaced by corresponding reactances. Therefore, for calculating the quantities Lef, gh , Lgh, ef , Lij, ef and Lij, gh , the expressions given in equations (5.41), (5.42), (5.54) and (5.57) have been used respectively, with the impedances replaced by the corresponding reactances. After obtaining the values of these four quantities, the factors LOSFM1 and LOSFM2 have been calculated from equations (5.63) and (5.64). The calculated values of these two factors for all the lines remaining in the system are shown in columns 5 and 6 of Table 5.3. Please note that for lines 3 and 17, thes two factors have no relevance and therefore, these values have been indicated as zero in this table for these two lines. After obtaining these two factors, the change in power flow in line ‘i-j’ are calculated from equation (5.62) using the information of pre-outage power flow in line no. 3 and 17. For ready reference, the pre-outage power flows of line 3 and 17 are given in columns 3 and 4 of Table 5.3 respectively. After calculating the change in power flow of any line ‘i-j’, its post-outage power flow is obtained by adding its pre-outage power flow to the change in power flow. Again, for ready reference, the pre-outage line power flows for all the lines are given in column 2 of Table 5.3. The calculated post-outage power flows in all the lines are given in column 7 of this table. Please note that, as line no. 3 and 17 are now out of the circuit, the post-outage power flows in these two lines are zero. Finally, for the purpose of comparison, complete AC power flow solution of the IEEE-14 bus system has been obtained after removing line no. 3 and 17 from the system. The line power flows obtained through AC power flow study are shown in the last column of Table 5.3. From the results given in last two columns it is observed that there is some difference between the power flows calculated by complete AC power flow analysis and contingency analysis. Moreover, comparison of Tables 5.2 and 5.3 shows that this difference is more for double line contingencies as compared to that for single line contingency.

5.6

Contingency ranking and selection

From the results of the Tables 5.1 and 5.2 it can be observed that the sensitivity factors give reasonably close estimates of real power flows in the lines in the event of outage of a generator or a line. However, the sensitivity factors give the estimate of only the real power flows over the lines. On the other hand, in several situations, it is also equally important to consider the bus 245

Table 5.3: Contingency calculation for outage of line no. 3 and 17 (all powers are given in p.u.) Line Pline P3 P17 LOSFM1 no. (ori) 1 1.6262 0.8655 0.0655 -0.2053 2 0.7464 0.8655 0.0655 0.2177 3 0.8655 0.8655 0.0655 0 4 0.5314 0.8655 0.0655 0.4758 5 0.3662 0.8655 0.0655 0.3491 6 -0.358 0.8655 0.0655 -1.072 7 -0.7068 0.8655 0.0655 -0.5508 8 0.2689 0.8655 0.0655 -0.0274 9 0.1063 0.8655 0.0655 -0.0107 10 0.2893 0.8655 0.0655 0.0238 11 0.1156 0.8655 0.0655 0.0237 12 0.0852 0.8655 0.0655 0 13 0.2005 0.8655 0.0655 0 14 0.0025 0.8655 0.0655 -0.0097 15 0.2664 0.8655 0.0655 -0.0177 16 0.0123 0.8655 0.0655 -0.0237 17 0.0655 0.8655 0.0655 0 18 -0.0777 0.8655 0.0655 -0.0237 19 0.0233 0.8655 0.0655 0 20 0.0857 0.8655 0.0655 0

LOSFM2 -0.025 0.0237 0 -0.0678 0.0389 -0.004 0.4444 -0.3998 -0.1579 0.5427 -0.5152 0.2174 0.7805 -0.01 -0.3898 0.507 0 0.5111 0.2304 0.9991

Pline Pline (cal) (ACLF) 1.44687535 1.5449 0.9363717 0.9864 0 0 0.938764 0.9803 0.670894 0.7059 -1.286078 -1.1902 -1.1544092 -1.1585 0.2189984 0.218 0.0866967 0.0862 0.34544575 0.3668 0.10236675 0.1236 0.0994397 0.0994 0.25162275 0.2558 -0.00655035 0.0023 0.22554875 0.2157 0.02499615 0.0069 0 0 -0.0647353 -0.0833 0.0383912 0.0373 0.15114105 0.1533

voltage variations as well as the reactive power flows over the lines in the event of any outage. In these situations, the full AC power flow analysis needs to be carried out as the sensitivity factors are not able to estimate the changes in bus voltage and the reactive power flows in the lines. However, the full AC power flow analysis is considerably slower than the sensitivity based methods and these are not suitable for analyzing thousands of potential outage cases within the time frame required by on-line contingency analysis. Thus, we have a contradicting situation here. On one hand, for fast evaluation of contingencies, sensitivity based methods need to be used whereas, for accurate estimation of the effects of any outage, slower, full AC power flow analysis needs to be carried out. To break this dilemma, a middle path is followed. Initially, all the outage studies are carried out using the sensitivity factors. Based on the results of the sensitivity analysis, all the outage cases are ranked according to a suitably chosen performance index (PI). Once the outage cases are ranked and sequentially arranged in decreasing order of the performance index, the top few outage cases are analyzed further in detail using the AC power flow analysis. Therefore, for contingency ranking, the choice of the performance index is very important. The PI should be such that it should satisfy the following criteria. a. It should adequately reflect the severity of any particular contingency. b. The final list of contingencies for which full AC power flow analysis is to be carried out, should 246

not be too short or too long. c. The PI should consider both real and reactive power variation in the system. To achieve the above objectives, different performance indices are used. Below, some of the most prominent performance indices used are discussed. The PI can be categorized into two groups; i) MW ranking method (in which the changes in real power flows only are considered) and ii) reactive power or voltage security ranking (in which the variations of voltage magnitude or reactive power only are considered). Of course, for considering both real and reactive power variations, the performance indices from these two categories need to be suitably combined. Now, let us look at the various indices from these two categories.

5.6.1

MW ranking methods

i) The simplest form of the PI is; L

P I = ∑ Wj [ j=i

n

Pj Pj max

]

(5.65)

In equation (5.65), ‘L’ is the number of lines in the system, Pj and Pj max are MW flow and MW capacity of the line ‘j’ respectively and ‘n’ is a suitable index. However, the PI given in equation (5.65) is prone to masking phenomenon in which a contingency causing many lines to be heavily loaded with no lines being overloaded is ranked higher than a contingency causing few lines to be overloaded with the remaining lines being lightly loaded. However, this masking phenomenon can be removed if the summation in equation (5.65) is taken over only the set of overloaded lines as shown in equation (5.66).

P I = ∑ Wj [ j∈SL

n

Pj Pj max

]

(5.66)

In equation (5.66), SL denotes the set of overloaded lines. Even with the PI of equation (5.66), there is a chance of misranking. For example, if a contingency C1 causes many lines to be slightly overloaded and another contingency C2 causes some lines to be heavily overloaded, then C2 is more severe than C1 . However, equation (5.66) may identify C1 to be more severe than C2 . To prevent this, a two term PI can be used as shown in equation (5.67).

P I = ∣Hd1 ∣ + ∑ Wj [ n1

j∈SL

n2

Pj Pj max

]

(5.67)

In equation (5.67), ∣Hd1 ∣ is the change in power flow in the highest overloaded line while n1 and n2 are suitable indices. In case the highest overload in two cases of contingency are same, then the effect of second highest overloaded line is also taken into account as shown in equation (5.68).

P I = ∣Hd1 ∣ + ∣Hd2 ∣ + ∑ Wj [ n1

n2

j∈SL

Pj Pj max

n3

]

(5.68)

In equation (5.68), ∣Hd2 ∣ denotes the change in power flow in the second highest overloaded 247

line. Similarly, if both the highest and second highest overloading conditions are same for two contingencies, then the third highest overloaded line is taken into account separately.

P I = ∣Hd1 ∣ + ∣Hd2 ∣ + ∣Hd3 ∣ + ∑ Wj [ n1

n2

n3

j∈SL

Pj Pj max

n4

]

(5.69)

As before,∣Hd3 ∣ denotes the change in power flow in third highest overloaded line. Of course, this same philosophy can be extended further to include more number of lines for the calculation of PI, if necessary.

5.6.2

Voltage security/reactive power ranking methods

These are several PIs suggested for properly ranking the voltage /reactive power contingencies. Some of them are: i) 2 αi ∆Vi ] P Iv = ∑ [ lim i=1 2 ∆Vi N

Where,

(5.70)

1 ∆Vi = Vi − Visp ; ∆Vilim = (Vimax − Vimin ) ; 2 Vi = post-contingency voltage magnitude of bus ‘i’; Visp = Nominal or specified voltage of bus ‘i’; Vimax , Vimin = Maximum and minimum limit of voltage magnitudes of bus ‘i’ αi = User selected weighting factor P Iv = Performance index corresponding to voltage security N = number of buses in the system

ii)

∣Vi − Vilim ∣ P Iv = ∑ Wvi Vilim i∈S1

(5.71)

where,

Vilim = Vimax if Vi > Vimax = Vimin if Vi < Vimin

(5.72)

Wvi in the weighting factor for bus ‘i’ and S1 in the set of all buses at which the voltage limits have been violated. iii)

P Iv = ∑ Wvi (∆Vi )2 i∈S1

where,

∆Vi =

∆Vinom − 1; ∆Vilim

∆Vinom = Vi − Vinom ; 248

(5.73)

1 Vinom = (Vimax + Vimin ); 2

1 ∆Vilim = (Vimax − Vimin ) 2

iv) 2 N dmax dmin i P Iv = ∑ [ max ] + ∑ [ imin ] i=1 ai i=1 ai N

2

(5.74)

Where,

Vi − V amax if Vi > Viamax Vinom = 0 otherwise

(5.75)

Vimin − Vi if Vi < Viamin nom Vi = 0 otherwise

(5.76)

dmax = i

dmin = i

amax = i

1 V

nom i

(Vimax − Viamax );

amin = i

1 Vinom

(Viamin − Vimin )

Viamax and Viamin are the higher and lower volatage alarm limits for bus ‘i’. v)

⎡ ⎤2 ⎢ ⎥ N ⎢ o nom ⎥ V + ∆V − V ⎢ ⎥ i i ⎥ P Iv = ∑ ⎢ i max min ⎢ ⎥ V − V i=1 ⎢ i i ⎥ ⎢ ⎥ 2 ⎣ ⎦

(5.77)

∆Vi is the post-contingency change in bus voltage magnitude of bus ‘i’. vi) m

N

1

m ∆Vi ∣ ] P Iv = [∑ Wi ∣ ∆Vimax i=1

(5.78)

The value of ‘m’ is taken to be very large (≈ 20) to avoid the masking effect. One popular way of combining the above discussed real power and reactive power ranking method is to use the 1P1Q method. In this method, a decoupled power flow is used and after one iteration, (one P-Q computation and one Q-V computation), the bus voltages are noted and with these bus voltages, the line power flows are calculated. It generally appears that there is sufficient information available (in the bus voltages) to arrive at a reasonable values of the performance indices. After calculating the real power and reactive power performance indices separately, the combined PI is calculated by adding the appropriate real power and reactive power indices together. With these descriptions of contingency ranking methods, we conclude our discussion of contingency analysis. From the next lecture, we will start our discussion of stability analysis.

249

Example of contingency analysis for two line outages

As an example, let us consider outage of two lines in the IEEE-14 bus system. Towards this goal, let us assume that line no. 3 (let it be considered as line ‘e-f’) and 17 (let it be considered as line ‘g-h’) have been taken out of the system. Subsequently, for any line ‘i-j’ (in the set of lines which are still remaining in the system), the factors LOSFM1 and LOSFM2 are calculated as follows.

LOSF M 1 =

Lij,ef + Lij,gh Lgh,ef 1 − Lef,gh Lgh,ef

(5.63)

LOSF M 2 =

Lij,gh + Lij,ef Lef,gh 1 − Lef,gh Lgh,ef

(5.64)

Now, under the assumption of DC power flow, for contingency calculation, the impedances are replaced by corresponding reactances. Therefore, for calculating the quantities Lef, gh , Lgh, ef , Lij, ef and Lij, gh , the expressions given in equations (5.41), (5.42), (5.54) and (5.57) have been used respectively, with the impedances replaced by the corresponding reactances. After obtaining the values of these four quantities, the factors LOSFM1 and LOSFM2 have been calculated from equations (5.63) and (5.64). The calculated values of these two factors for all the lines remaining in the system are shown in columns 5 and 6 of Table 5.3. Please note that for lines 3 and 17, thes two factors have no relevance and therefore, these values have been indicated as zero in this table for these two lines. After obtaining these two factors, the change in power flow in line ‘i-j’ are calculated from equation (5.62) using the information of pre-outage power flow in line no. 3 and 17. For ready reference, the pre-outage power flows of line 3 and 17 are given in columns 3 and 4 of Table 5.3 respectively. After calculating the change in power flow of any line ‘i-j’, its post-outage power flow is obtained by adding its pre-outage power flow to the change in power flow. Again, for ready reference, the pre-outage line power flows for all the lines are given in column 2 of Table 5.3. The calculated post-outage power flows in all the lines are given in column 7 of this table. Please note that, as line no. 3 and 17 are now out of the circuit, the post-outage power flows in these two lines are zero. Finally, for the purpose of comparison, complete AC power flow solution of the IEEE-14 bus system has been obtained after removing line no. 3 and 17 from the system. The line power flows obtained through AC power flow study are shown in the last column of Table 5.3. From the results given in last two columns it is observed that there is some difference between the power flows calculated by complete AC power flow analysis and contingency analysis. Moreover, comparison of Tables 5.2 and 5.3 shows that this difference is more for double line contingencies as compared to that for single line contingency.

5.6

Contingency ranking and selection

From the results of the Tables 5.1 and 5.2 it can be observed that the sensitivity factors give reasonably close estimates of real power flows in the lines in the event of outage of a generator or a line. However, the sensitivity factors give the estimate of only the real power flows over the lines. On the other hand, in several situations, it is also equally important to consider the bus 245

Table 5.3: Contingency calculation for outage of line no. 3 and 17 (all powers are given in p.u.) Line Pline P3 P17 LOSFM1 no. (ori) 1 1.6262 0.8655 0.0655 -0.2053 2 0.7464 0.8655 0.0655 0.2177 3 0.8655 0.8655 0.0655 0 4 0.5314 0.8655 0.0655 0.4758 5 0.3662 0.8655 0.0655 0.3491 6 -0.358 0.8655 0.0655 -1.072 7 -0.7068 0.8655 0.0655 -0.5508 8 0.2689 0.8655 0.0655 -0.0274 9 0.1063 0.8655 0.0655 -0.0107 10 0.2893 0.8655 0.0655 0.0238 11 0.1156 0.8655 0.0655 0.0237 12 0.0852 0.8655 0.0655 0 13 0.2005 0.8655 0.0655 0 14 0.0025 0.8655 0.0655 -0.0097 15 0.2664 0.8655 0.0655 -0.0177 16 0.0123 0.8655 0.0655 -0.0237 17 0.0655 0.8655 0.0655 0 18 -0.0777 0.8655 0.0655 -0.0237 19 0.0233 0.8655 0.0655 0 20 0.0857 0.8655 0.0655 0

LOSFM2 -0.025 0.0237 0 -0.0678 0.0389 -0.004 0.4444 -0.3998 -0.1579 0.5427 -0.5152 0.2174 0.7805 -0.01 -0.3898 0.507 0 0.5111 0.2304 0.9991

Pline Pline (cal) (ACLF) 1.44687535 1.5449 0.9363717 0.9864 0 0 0.938764 0.9803 0.670894 0.7059 -1.286078 -1.1902 -1.1544092 -1.1585 0.2189984 0.218 0.0866967 0.0862 0.34544575 0.3668 0.10236675 0.1236 0.0994397 0.0994 0.25162275 0.2558 -0.00655035 0.0023 0.22554875 0.2157 0.02499615 0.0069 0 0 -0.0647353 -0.0833 0.0383912 0.0373 0.15114105 0.1533

voltage variations as well as the reactive power flows over the lines in the event of any outage. In these situations, the full AC power flow analysis needs to be carried out as the sensitivity factors are not able to estimate the changes in bus voltage and the reactive power flows in the lines. However, the full AC power flow analysis is considerably slower than the sensitivity based methods and these are not suitable for analyzing thousands of potential outage cases within the time frame required by on-line contingency analysis. Thus, we have a contradicting situation here. On one hand, for fast evaluation of contingencies, sensitivity based methods need to be used whereas, for accurate estimation of the effects of any outage, slower, full AC power flow analysis needs to be carried out. To break this dilemma, a middle path is followed. Initially, all the outage studies are carried out using the sensitivity factors. Based on the results of the sensitivity analysis, all the outage cases are ranked according to a suitably chosen performance index (PI). Once the outage cases are ranked and sequentially arranged in decreasing order of the performance index, the top few outage cases are analyzed further in detail using the AC power flow analysis. Therefore, for contingency ranking, the choice of the performance index is very important. The PI should be such that it should satisfy the following criteria. a. It should adequately reflect the severity of any particular contingency. b. The final list of contingencies for which full AC power flow analysis is to be carried out, should 246

not be too short or too long. c. The PI should consider both real and reactive power variation in the system. To achieve the above objectives, different performance indices are used. Below, some of the most prominent performance indices used are discussed. The PI can be categorized into two groups; i) MW ranking method (in which the changes in real power flows only are considered) and ii) reactive power or voltage security ranking (in which the variations of voltage magnitude or reactive power only are considered). Of course, for considering both real and reactive power variations, the performance indices from these two categories need to be suitably combined. Now, let us look at the various indices from these two categories.

5.6.1

MW ranking methods

i) The simplest form of the PI is; L

P I = ∑ Wj [ j=i

n

Pj Pj max

]

(5.65)

In equation (5.65), ‘L’ is the number of lines in the system, Pj and Pj max are MW flow and MW capacity of the line ‘j’ respectively and ‘n’ is a suitable index. However, the PI given in equation (5.65) is prone to masking phenomenon in which a contingency causing many lines to be heavily loaded with no lines being overloaded is ranked higher than a contingency causing few lines to be overloaded with the remaining lines being lightly loaded. However, this masking phenomenon can be removed if the summation in equation (5.65) is taken over only the set of overloaded lines as shown in equation (5.66).

P I = ∑ Wj [ j∈SL

n

Pj Pj max

]

(5.66)

In equation (5.66), SL denotes the set of overloaded lines. Even with the PI of equation (5.66), there is a chance of misranking. For example, if a contingency C1 causes many lines to be slightly overloaded and another contingency C2 causes some lines to be heavily overloaded, then C2 is more severe than C1 . However, equation (5.66) may identify C1 to be more severe than C2 . To prevent this, a two term PI can be used as shown in equation (5.67).

P I = ∣Hd1 ∣ + ∑ Wj [ n1

j∈SL

n2

Pj Pj max

]

(5.67)

In equation (5.67), ∣Hd1 ∣ is the change in power flow in the highest overloaded line while n1 and n2 are suitable indices. In case the highest overload in two cases of contingency are same, then the effect of second highest overloaded line is also taken into account as shown in equation (5.68).

P I = ∣Hd1 ∣ + ∣Hd2 ∣ + ∑ Wj [ n1

n2

j∈SL

Pj Pj max

n3

]

(5.68)

In equation (5.68), ∣Hd2 ∣ denotes the change in power flow in the second highest overloaded 247

line. Similarly, if both the highest and second highest overloading conditions are same for two contingencies, then the third highest overloaded line is taken into account separately.

P I = ∣Hd1 ∣ + ∣Hd2 ∣ + ∣Hd3 ∣ + ∑ Wj [ n1

n2

n3

j∈SL

Pj Pj max

n4

]

(5.69)

As before,∣Hd3 ∣ denotes the change in power flow in third highest overloaded line. Of course, this same philosophy can be extended further to include more number of lines for the calculation of PI, if necessary.

5.6.2

Voltage security/reactive power ranking methods

These are several PIs suggested for properly ranking the voltage /reactive power contingencies. Some of them are: i) 2 αi ∆Vi ] P Iv = ∑ [ lim i=1 2 ∆Vi N

Where,

(5.70)

1 ∆Vi = Vi − Visp ; ∆Vilim = (Vimax − Vimin ) ; 2 Vi = post-contingency voltage magnitude of bus ‘i’; Visp = Nominal or specified voltage of bus ‘i’; Vimax , Vimin = Maximum and minimum limit of voltage magnitudes of bus ‘i’ αi = User selected weighting factor P Iv = Performance index corresponding to voltage security N = number of buses in the system

ii)

∣Vi − Vilim ∣ P Iv = ∑ Wvi Vilim i∈S1

(5.71)

where,

Vilim = Vimax if Vi > Vimax = Vimin if Vi < Vimin

(5.72)

Wvi in the weighting factor for bus ‘i’ and S1 in the set of all buses at which the voltage limits have been violated. iii)

P Iv = ∑ Wvi (∆Vi )2 i∈S1

where,

∆Vi =

∆Vinom − 1; ∆Vilim

∆Vinom = Vi − Vinom ; 248

(5.73)

1 Vinom = (Vimax + Vimin ); 2

1 ∆Vilim = (Vimax − Vimin ) 2

iv) 2 N dmax dmin i P Iv = ∑ [ max ] + ∑ [ imin ] i=1 ai i=1 ai N

2

(5.74)

Where,

Vi − V amax if Vi > Viamax Vinom = 0 otherwise

(5.75)

Vimin − Vi if Vi < Viamin nom Vi = 0 otherwise

(5.76)

dmax = i

dmin = i

amax = i

1 V

nom i

(Vimax − Viamax );

amin = i

1 Vinom

(Viamin − Vimin )

Viamax and Viamin are the higher and lower volatage alarm limits for bus ‘i’. v)

⎡ ⎤2 ⎢ ⎥ N ⎢ o nom ⎥ V + ∆V − V ⎢ ⎥ i i ⎥ P Iv = ∑ ⎢ i max min ⎢ ⎥ V − V i=1 ⎢ i i ⎥ ⎢ ⎥ 2 ⎣ ⎦

(5.77)

∆Vi is the post-contingency change in bus voltage magnitude of bus ‘i’. vi) m

N

1

m ∆Vi ∣ ] P Iv = [∑ Wi ∣ ∆Vimax i=1

(5.78)

The value of ‘m’ is taken to be very large (≈ 20) to avoid the masking effect. One popular way of combining the above discussed real power and reactive power ranking method is to use the 1P1Q method. In this method, a decoupled power flow is used and after one iteration, (one P-Q computation and one Q-V computation), the bus voltages are noted and with these bus voltages, the line power flows are calculated. It generally appears that there is sufficient information available (in the bus voltages) to arrive at a reasonable values of the performance indices. After calculating the real power and reactive power performance indices separately, the combined PI is calculated by adding the appropriate real power and reactive power indices together. With these descriptions of contingency ranking methods, we conclude our discussion of contingency analysis. From the next lecture, we will start our discussion of stability analysis.

249