PART III ADVANCED TOPICS IN POWER SYSTEM DYNAMICS
14.6 Properties of Coherency-Based Equivalents
14.6.4 Eigenvalues and Eigenvectors of the Equivalent Model
The analysis in the previous subsection was undertaken under an assumption that, in the state equation (14.83), changes of power in the remaining part of the system constitute a disturbance.
Such a model was used to investigate internal group swings and external swings between the group and the rest of the system. The model could not be used to asses the influence of aggregation of nodes in group{A}on the modes corresponding to oscillations in the rest of the system. That task will require the creation of the incremental model of the whole system and an investigation of how aggregation of group{A}influences eigenvalues and eigenvectors in the whole system. This difficult task will be simplified by reducing the system model using aggregation which will be shown as a projection of the state space on a subspace.
Letxbe the state vector of a dynamic system described by the state equation
˙
x= Ax. (14.103)
System reduction will be undertaken by projecting vectorxonto a smaller vector
xe=Cx, (14.104)
whereCis a rectangular matrix defining this projection and further referred to as theprojection matrix. The lower index comes from the word ‘equivalent’. The reduced model is described by
˙
xe=axe, (14.105)
whereais a square matrix that will now be expressed using matricesAandC.
Equation (14.105) describes a reduced dynamic system obtained by the reduction of the state vector using transformation (14.104).
Differentiating both sides of Equation (14.104) gives ˙xe=Cx. Substitution of ˙˙ xeby the right hand side of Equation (14.105) leads toaxe=CAx. Substitution ofxeby the right hand side of (14.104) givesaCx=C Axwhich finally leads to
aC=CA. (14.106)
Right-multiplying byCTgivesaCCT=C ACTleading to
a=CACT(CCT)−1, (14.107)
where matrixCCTis a square matrix with rank equal to the number of state variables in the reduced model.
The relationship given by (14.106) is very important because it will make it possible to show that the reduced model (14.105) obtained from reducing the state vector using transformation (14.104) partially retains eigenvalues and eigenvectors of the original (unreduced) system (14.103).
Let λi be an eigenvalue of the state matrix A in Equation (14.103) and let wi be a right eigenvector of that matrix. Then according to the definition of eigenvectors, Awi =λiwi. Left- multiplying byCgivesC Awi =λiCwi. Substitution ofCAby the left hand side of (14.106) results in aCwi=λiCwior
awei =λiwei, (14.108)
where
wei =Cwi. (14.109)
Equation (14.108) shows that for eachwei =0the numberλiis an eigenvalue of matrixaandwei
is the corresponding right eigenvector. Obviouslyλi is also an eigenvalue ofA. Equation (14.109) shows that vectorwei is created by the reduction of vectorwi. This means that by satisfying the condition
wei=Cwi =0, (14.110)
the reduced dynamic system (14.105) obtained by reducing the state vector using (14.104) partially retains eigenvalues and eigenvectors of the original (unreduced) system (14.103). Note that the relationship between eigenvector wei of the reduced model and eigenvector wi of the original (unreduced) model is the same as that between the state vectorxeand the state vectorx. This means thatweicorresponds to the projection ofwiobtained using the projection matrixC.
Obviously condition (14.110) is not satisfied for every matrixCand the reduced model does not maintain all eigenvalues and eigenvectors of the original (unreduced) model.
In the case of the incremental model of a power system, Machowski (1985) showed that the projection matrix should be of the following form:
C=
1
. .. 0
- - - -1
0 1
n 1 n ã ã ã 1
n
=
1 0
- - - -
0 1
n1TA
, (14.111)
where1Ais a vector of ones andnis the number of generators in group{A}.
The discussed reduction using the projection matrix may be applied to Equation (14.84) or (14.83). This will be shown for the latter since: (i) the state matrix in (14.83) is simpler than in (14.84); (ii) there is an exact relationshipλi = √àibetween the eigenvalues of both matrices.
When applying the projection matrix (14.111), vectorδin Equation (14.62) is transformed in the following way:
C δR
δA
= δR
δa
, (14.112)
where
δa = 1 n
j∈{A}
δj. (14.113)
Equation (14.113) shows that, when using the discussed reduction method, the rotor of the equivalent generator moves on average with respect to all the rotors of aggregated generators.
Obviously, for exactly coherent generators this movement is the same for all the generators in group {A}and its average value is equal simply to the value for each generator in the group.
For matrixCin the structure (14.111) it can be shown that
CT(CCT)−1=
1
. .. 0A
- - - -1
0RA
1 1 ... 1
=
1 0A
- - - - 0RA 1A
=BT. (14.114)
Thus the state matrix of the reduced model given by Equation (14.107) takes the simple form
a=CABT. (14.115)
The matrix equation of motion, with damping neglected, for the original model (Figure 14.6a) containing generators in groups{A}and{R}can be written similarly to Equation (11.23):
δ¨R
- - - δ¨A
= −
M−R1HRR M−R1HRA
- - - - M−A1HAR M−A1HAA
δR
- - - - δA
. (14.116)
After applying reduction using matrixCin the form (14.111), the state vector is reduced to the form (14.112) while Equation (14.116) reduces to
δ¨R
- - - δ¨a
= −
M−R1HRR M−R1HRA1A
- - - - 1
n1TAM−A1HAR
1
n1TAM−A1HAA1A
δR
- - - - δa
, (14.117)
where the state matrix has been calculated according to (14.115). As shown previously, the reduced model given by (14.117) partially retains eigenvalues and eigenvectors of the original (unreduced) model of (14.116).
It is easy to see some similarity between the described reduced model (14.117) and the reduced model obtained using the Di Caprio and Marconato aggregation described in Section 14.6.2. In both cases there is a summation of matrix elements corresponding to multiplication by1Aand1TA. Using Equation (14.62) obtained from the Di Caprio and Marconato aggregation. it is possible, as in (14.117), to write the following state equation:
δ¨R
- - - δ¨a
= −
M−R1HRR M−R1HRA1A
- - - - Ma−11TAHAR Ma−11TAHAA1A
δR
- - - - δa
, (14.118)
where, according to (14.33), the inertia coefficients of the equivalent machine areMa=
i∈{A}Mi. It is also easy to see, when comparing Equations (14.117) and (14.118), that they differ in the bottom row corresponding to the equivalent generator. The difference lies in the different order of factors, which is important for the result as the multiplication of matrices is generally not commutative. A detailed analysis leads to the conclusion that the elements in the bottom row of Equation (14.117) are given by
aak= −1 n
i∈{A}
Hi k
Mi
, (14.119)
and those in Equation (14.118) are given by
aak= −
i∈{A}Hi k
i∈{A}Mi
. (14.120)
This is obvious because generally both elements given by Equations (14.119) and (14.120) are not the same. In the particular case when Equation (14.39) is satisfied, that is when the group is exactly coherent, the following holds:
Hi k
Mi
=hk for i,j∈ {A}, k∈ {B}. (14.121) HenceHi k=hkMi. Substituting this into (14.120) gives
aak= −
i∈{A}hkMi
i∈{A}Mi
= − hk
i∈{A}Mi
i∈{A}Mi
= −hk. (14.122)
The same value ofaak= −hkcan be obtained by substituting (14.121) into (14.119). This shows that when the exact coherency condition (14.39) is satisfied, the matrices in (14.117) and (14.118) are the same.
Example 14.2
To illustrate how the reduced model partially retains eigenvalues and eigenvectors, a simple three- machine system will be studied in which two generators satisfy the exact coherency condition given by (14.121). The state matrix given by Equation (14.116) is
−6 3 3 - - - -
2 −4 2
2 3 −5
.
The eigenvalues and eigenvectors are
à1=0 and w1= 1 1 1T
à2= −8 and w2= −3 1 1T
à3= −7 and w3= 3−4 3T
. The state matrix reduces using Equation (14.117) to
−6 6 - - - -
2 −2
.
The eigenvalues and eigenvectors of this state matrix are à1=0 and we1= 1 1T
à2= −8 and we2= −3 1T
.
The reduced system also retained, apart from the zero eigenvalue, the eigenvalueà2= −8 and the associated right eigenvectorwe2= −3 1T
which is a part of the original eigenvector w2= −3 1 1T
. Equation (14.110) is satisfied as
Cw2=
1 0 0 - - - -
0 1
2 1 2
−3 1 1
= −3
1
=we2.
This illustrates that the reduced model partially retains eigenvalues and eigenvectors of the original (unreduced) model.
To summarize the observations contained in this chapter:
1. The operations of aggregation and linearization are commutative (proof in Section 14.6.2).
2. The reduced linear model (14.62) obtained using the method of Di Caprio and Marconato corresponds to the linearized form of the reduced model obtained by Zhukov’s aggregation (proof in Section 14.6.2).
3. When the exact coherency condition given by Equation (14.39) is satisfied, the reduced lin- ear model (14.118) is equivalent to the reduced model (14.117) obtained using transformation (14.111) and the projection matrix (14.111).
4. The reduced model (14.117) partially retains the eigenvalues of the original (unreduced) model.
These observations clearly show that, when the exact coherency condition (14.39) is satisfied, the reduced model obtained by Zhukov’s aggregation (Section 14.2.3) also partially retains the eigenvalues of the original (unreduced) model. This is a very important property of the coherency- based dynamic equivalent model obtained by Zhukov’s aggregation.
In practice, exact coherency rarely occurs in real power systems apart from identical generators operating on the same busbar. Reduced dynamic models are created by aggregation of generators for which the coherency definition is satisfied within accuracyεδas in condition (14.29). Obviously any inaccuracy of coherency means that all the dynamic properties of the original (unreduced) model will be maintained only to some degree by the equivalent (reduced) model. Hence it may be expected that also eigenvalues and eigenvectors of the equivalent (reduced) model will be only approximately equal to eigenvalues and eigenvectors of the original (unreduced) model. It is important here that the equivalent (reduced) model maintains as precisely as possible those modal variables that are strongly excited by disturbances in the internal subsystem and which therefore have the strongest influence on power swings in the internal subsystem. These modal variables will be referred to asdominant modal variables(see also Section 12.1.6). Modal analysis (Section 12.1) shows that matricesUandWbuilt from right and left eigenvectors decide which modal variables are most strongly excited and influence power swings. The example below will show that a coherency-based equivalent model quite accurately retains the dominant modes.
Example 14.3
Figure 14.17 shows a 15-machine test system. Plant 7 was assumed to constitute the internal system. For this internal system, the algorithm described in Section 14.5 was used to identify coherent groups which are encircled in Figure 14.17 using solid lines.
1 2
4 6
5
8 3
10 11
12 9
13
14 1 15
2
3 4
5 6
7
8
9 10
11
12 13
14 15
26
25 22
20
18 19
23 24
17
28
16
27 21
29 7
Figure 14.17 Test system and recognized coherent groups.
The dominant modes have been identified assuming that the initial disturbance is a rotor angle change of generator 7, that isδ= 0ã ã ã0|δ7|0ã ã ã0T
. With this disturbance, the equation z=Uãδresults inz=u◦7ãδ7whereu◦7denotes the seventh column of matrixU. For the assumed data (Machowskiet al., 1986; Machowski, Gubina and Omahen, 1986), the following results were obtained:
rfor the original (unreduced) model
u◦7=10−3ã −184 0 914 −11 160 −3 −56 −9 −71 −18 −64 −1 −90 0 20T
;
rfor the equivalent (reduced) model
u◦7=10−3ã −154 29 915 −47 124 −2 0T
.
The largest values correspond to the third modal variable z3 and are shown in bold and underlined. Note that they are almost the same for the original and the equivalent model, which means that excitation of the third modal variable in both systems is the same. Also strongly exited are the first modal variablez1and the fifthz5. That excitation is nevertheless several times weaker than excitation of the third modal variablez3. The remaining values are much smaller, so it may be assumed that the remaining modes are either weakly excited or not excited at all. The excited modal variables are associated with the following eigenvalues:
original model : equivalent model : à1= −11.977 à1= −13.817 à3= −42.743 à3= −44.170 à5= −72.499 à5= −119.390.
Clearly the third eigenvalue corresponding to the most excited modal variable is almost the same for both the equivalent and original models. The first eigenvalue has similar values for both models while the fifth is quite different. However, it should be remembered that the first and fifth modal variables are weakly excited and do not have to be accurately modelled.
MatrixW decides how individual modal variables influence power swings in the internal subsystem. The equationzδ=Wzresults inδ7=w7◦zwherew7◦denotes the seventh row ofW. For the assumed data the following results were obtained:
rfor the original (unreduced) model
w7◦=10−1ã −34 −26 98 −10 21 0 −50 −10 −80 −20 −40 0 0 0 −20
;
rfor the equivalent (reduced) model
w7◦=10−1ã −52 6 99 −12 60 0 7 .
The largest values again correspond to the third modal variablez3 and they are almost the same for both models. This means that the influence of the third modal variable on power swings in the internal system is the same in both models. The values for the first modal variablez1
and the fifthz5 are quite different, but those modal variables are weakly excited. Nevertheless, model reduction by aggregation causes some differences between power swings simulated in both models – see the simulation results shown previously in Figure 14.14 for a different test system.
By making use of u◦7 (seventh column of U) and w7◦ (seventh row of W) it is possible to calculate participation factors defined in Section 11.1. According to Equation (12.90), it is necessary to multiply elements of matrix columnu◦7by elements of matrix roww7◦. For example, the first participation factor for the original (unreduced) model is: 10−4ã184ã34∼=63ã10−2. The calculated participation factors can be expressed in the following way:
rfor the original (unreduced) model
10−2ã 63 0 896 1 34 0 28 1 57 4 26 0 0 0 −4
;
rfor the equivalent (reduced) model
10−2ã 80 2 906 6 74 0 0 .
Based on the values of participation factors, it can be concluded that there is a strong relation- ship between the investigated variableδ7in the internal subsystem and the third modal variable z3. The relationships betweenδ7 and the first modal variablez1and the fifthz5are an order of magnitude weaker.
When analysing Example 14.3 it should be remembered that the calculated eigenvaluesài are the eigenvalues of a matrix in the second-order equation, respectively (14.116) and (14.117). These values are real and negative. The corresponding eigenvaluesλi of first-order equations of the type (14.84) are complex numbersλi = √ài.