PART III ADVANCED TOPICS IN POWER SYSTEM DYNAMICS
14.6 Properties of Coherency-Based Equivalents
14.6.5 Equilibrium Points of the Equivalent Model
The coherency-based equivalent model obtained by Zhukov’s aggregation is constructed for a stable equilibrium point which is at the same time the steady-state operating point of the system.
Consequently, the equivalent model must partially retain the coordinates of the stable equilibrium.
This can be illustrated in the following way when denoting the nodes as in Figure 14.6. Letrbe the number of generators in group{R}andN be the total number of system generators, that is in both groups{R}and{A}. Then the coordinates of the stable equilibrium point of the original (unreduced) model and the equivalent (reduced) model can be written as
δˆ= δˆ1 ã ã ã δˆr δˆr+1 ã ã ã δˆN
T
, (14.123)
δˆe= δˆ1 ã ã ã δˆr δˆa
T
, (14.124)
where ˆδais the power angle of the equivalent generator given by Equation (14.22). Now the question arises whether, and which, unstable equilibrium points are retained by the reduced (equivalent) model. This question is especially important from the point of view of the Lyapunov direct method.
It was shown in Section 6.3.5 (Figure 6.24) that when transient stability is lost, each unstable equilibrium point corresponds to the system splitting in a certain way into groups of asynchronously operating generators. From that point of view the reduced (equivalent) model is a good model if it partially retains those unstable equilibrium points which are important for disturbances in the internal subsystem (Figure 14.1).
The coordinates of an unstable equilibrium point of the original (unreduced) model and the equivalent (reduced) model will be denoted as follows:
δ˜= δ˜1 ã ã ã δ˜r δ˜r+1 ã ã ã δ˜N
T
, (14.125)
δ˜e= δ˜e1 ã ã ã δ˜er δ˜a
T
. (14.126)
The equivalent model will be said to partially retain the unstable equilibrium point of the original model if
δ˜ek =δ˜k for k∈ {R}. (14.127) The electrical interpretation of Zhukov’s aggregation shown in Figure 14.15 will reveal which particular unstable equilibrium points satisfy Equation (14.127). Aggregation will not distort the coordinates of an unstable equilibrium point if at that point the ratio of the voltages is equal to the transformation ratio used for aggregation, that is the ratio of voltages at the stable equilibrium point. As in Equation (14.16), the condition may be written as
V˜−a1V˜A=ϑ=Vˆ−a1VˆA. (14.128)
For the classical generator model (constant magnitudes of emfs) the condition simplifies to δ˜i−δ˜a=δˆi−δˆa for i∈ {A}, (14.129) or ˜δi−δˆi=δ˜a−δˆa. This equation must be satisfied for eachi∈ {A}and therefore for eachi,j∈ {A}.
Hence ˜δi−δˆi =δ˜j−δˆj =δ˜a−δˆa must be satisfied, or
δ˜i−δˆi =δ˜j−δˆj i,j∈ {A}. (14.130) This means that for each generator belonging to a given groupi,j∈ {A}, the distance between an unstable equilibrium point and the stable equilibrium point must be the same. Such unstable equilibrium points can be calledpartially equidistant pointswith respect to a given group of variables belonging to group{A}.
The equivalent model obtained using Zhukov’s aggregation partially retains each unstable equi- librium point equidistant with respect to a given group of variables belonging to group {A}.
Aggregation destroys only those unstable equilibrium points that are not partially equidistant. This property will be illustrated using an example that is intuitively simple to understand.
Example 14.4
Figure 14.18 shows an example of two parallel generators 1 and 2 operating on an infinite busbar represented by a generator of large capacity 3. For each external short circuit in the transmission line 4–3, the two parallel generators are exactly coherent. Oscillations between the generators may appear only in the case of an internal short circuit inside the power plants at nodes 5 or 6.
The lower part of Figure 14.18a shows the equivalent diagram after elimination of load nodes.
The parameters have symbols following the notation in Equation (6.41). Figure 14.18b shows equiscalar lines of potential energy similar to Figure 6.24.
1 2
5 6
4 7
3
b12= 0.2
b13= 1.0
b23= 1.0 Pm1–P01= 0.5
Pm2–P02= 0.5
Pm3–P03= –1.0 1
2
3
(a) (b)
3 –1
1 2 3
–1 –2
2 1 s
1.3 1.2 1.0 0.8 0.3
1.00.8 1.5 3.53.0 4.0 5.0 A
B δ′23
δ′13 u1 u2
u3 0.98
0.98 1.37
–2
Figure 14.18 Illustration to the definition of the partially equidistant equilibrium point: (a) network diagrams; (b) equiscalar lines of potential energy.
There are three unstable equilibrium points: u1, u2, u3. The saddle point u1 corresponds to the loss of synchronism of generator 1 with respect to generators 2 and 3. This may happen when a short circuit appears at node 5. The saddle point u2 corresponds to a loss of synchronism of generator 2 with respect to generators 1 and 3. This may happen when there is a short circuit at
node 6. Point u3 is of the maximum type. It corresponds to a loss of synchronism of generators 1 and 2 with respect to generator 3. This may happen when there is a short circuit in line 4–3 at, for example at point 7. For point u3, condition (14.130) is satisfied as ˜δ13−δˆ13=δ˜23 −δˆ23 . Point u3 is at the same time partially equidistant. Note that when the exact coherency condition is satisfied, trajectoryδ(t) lies on the straight line AB crossing the origin, point s and point u3.
The line is defined by
δ13 (t)−δ23 (t)=δˆ13 −δˆ23 =δˆ12=constant,
similar to Equation (14.31). Aggregation of generators 1 and 2 reduces the three-machine system to a two-machine system and destroys the unstable equilibrium points u1, u2. After aggregation the unstable equilibrium point u3 is retained. A plot of potential energy for the reduced model (two-machine model) corresponds to a cross-section of the diagram in Figure 14.18 along line AB. This plot has the same shape as shown previously in Figure 6.21b.
The next important issue for the Lyapunov direct method is the question whether or not the dynamic equivalent (reduced) model retains the values of the Lyapunov function during the transient state and at unstable equilibrium points of the original (unreduced) model. For the Lyapunov function V(δ, ω)=Ek+Ep given by Equation (6.52) the answer to this question is positive, which will now be proved.
For kinetic energyEkthe proof is trivial. It is enough to separate Equation (6.46) into two sums:
Ek= 1 2
N i=1
Miω2i =1 2
i∈{R}
Miωi2+1 2
i∈{A}
Miω2i = 1 2
i∈{R}
Miωi2+1 2Maω2a, where fori∈ {A}the definition of exact coherency givesω1= ã ã ã =ωn=ωaand, according to Equation (14.33),Ma=
i∈{A}Mi. This concludes the proof.
For potential energy given by Equation (6.51) the proof is also simple but long. Here only an outline will be given:
1. The sum of components (Pmi−P0i)(δi−δˆi) should be broken down (similarly as for kinetic energy) into two sums: one fori∈ {R}and one fori∈ {A}. Then it should be noted that when the exact coherency condition is satisfied fori∈ {A}, then (δi−δˆi)=(δa−δˆa) while, according to the principles of aggregation,
i∈(A)(Pmi−P0i)=(Pma−P0a).
2. The double sum of componentsbi j(cosδi j −cos ˆδi j) in Equation (6.51) should be broken down into three sums: (i) fori,j∈ {R}; (ii) fori∈ {R}, j∈ {A}; and (iii) fori,j∈ {A}. Then it should be noted that componentsbi jcosδi j andbi jcos ˆδi j correspond to synchronizing powers. It was shown in Section 14.6.2 that for the equivalent (reduced) model, synchronizing powers are equal to the sum of synchronizing powers of aggregated generators. Hence the corresponding sums of components give the same values as for the equivalent (reduced) model.
Conclusions from the above points 1 and 2 conclude the proof for potential energy. This will now be illustrated using the results of calculations conducted for a test system.
Example 14.5
Consider again the test system shown in Figure 14.17. In this example the internal subsystem is assumed to consist of power plant 11 located in the middle of the test network. Treating the test system as the original (unreduced) model, the gradient method was used to calculate the coordinates of the stable equilibrium point and the unstable equilibrium point corresponding to the loss of synchronism of generator 11. The coordinates of those points are shown in Table
14.1 in columns under the heading ‘Original’. For the assumed internal subsystem, the coherency recognition algorithm has identified two groups:{2, 3, 4}and{5, 6, 8, 9, 10, 12, 13, 14, 15}.
The groups have been aggregated using Zhukov’s method. For the equivalent (reduced) model obtained, the stable equilibrium point and unstable equilibrium point corresponding to the loss of synchronism of generator 11 have been calculated. The coordinates of these points are shown in Table 14.1 in columns under the heading ‘Reduced’. The results show that for generators{1, 7, 11}, the coordinates for both the stable and unstable equilibrium points have been well retained.
The lower rows of Table 14.1 show the values of the Lyapunov function calculated for the unstable equilibrium point of the original (unreduced) and equivalent (reduced) model. Clearly the values are quite close, similar to the values of the critical clearing time for a short circuit in busbar 11.
Table 14.1 Results for a fault at bus 11
Coordinates of equilibrium points
Stable Unstable
Generator no. Group no. Original Reduced Original Reduced
1 — 0.00 0.00 0.00 0.00
7 — 23.36 23.40 50.76 49.65
11 — 14.22 14.30 183.80 181.81
2 1 20.54 19.65 26.42 24.50
3 19.84 25.10
4 10.56 19.02
5 2 13.25 18.24 28.22 34.68
6 12.48 27.02
8 15.39 26.58
9 12.73 28.28
10 11.15 26.59
12 14.23 33.02
13 14.14 34.44
14 31.08 52.63
15 25.55 44.67
Value of Lyapunov function 11.05 10.95
Critical clearing time 0.322 0.325
Similar results have been obtained for the same and other test systems when choosing different internal subsystems. More examples can be found in the publications by Machowski (1985) and Machowskiet al.(1986, 1988).
Appendix