Dynamical systems possess some uniquely nonlinear phenomena, such as local bifur-
cations [92]. A vector field f (x, µ) is said to undergo a fixed point bifurcation when
the flow around a fixed point x0 changes qualitatively, when a parameter µ crosses
some critical value µ0.
Local bifurcations are very important in natural and engineered systems. For example, in power systems it has been argued that the significant practical problem of voltage collapse in fact has its origin in a saddle-node bifurcation, where the operating equilibrium point suddenly disappears as a consequence of a change in the parameters (for example, reactive load). From a practical viewpoint, is it absolutely critical to recognize such situations, and choose nominal values for the parameters that are far away from the hypersurface where bifurcations occur.
Despite its practical importance, there does not seem to be many systematic approaches to the problem of computing bifurcation margins. In reference [30], Dobson proposed two methods for computing locally closest bifurcations to a given set of nominal parameters. These methods (iterative and direct) aim to numerically solve the equations characterizing the closest point in the bifurcation surface. How- ever, the problem with this approach is exactly the same as in standard robustness analysis: what we really need in practice is some way of guaranteeing a minimum
distance (or safety margin) to a singularity, not just feasible solutions. In other
words, if we find a bifurcation “nearby,” then we need to be absolutely sure that there are no other points that are even closer. The results in [30, 2] do not fully address this issue: a Monte Carlo approach is employed, where the optimization is restarted from multiple initial conditions.
The techniques developed in previous chapters can be applied to rigorously prove bounds on the distance to the bifurcation surface. The conditions for a vector field
f (x, µ) to have a saddle-node bifurcation at (x0, µ0) are [38]:
f = 0 w∗Dxf = 0
w∗Dµf 6= 0
where v, w are the right and left eigenvectors, respectively, of the jacobian J := Dxf ,
corresponding to the simple eigenvalue zero. The two conditions on the left-hand side correspond to the singularity of the jacobian at the fixed point, and the ones on the right-hand side are generic transversality requirements.
As we can see, in the polynomial (or rational) case, the set where bifurcations occur is semialgebraic, since it can be characterized in the form described by The- orem 4.4. Therefore, our methods are immediately applicable to this problem.
The example below also demonstrates another issue: even if the problem con- tains nonalgebraic elements, such as trigonometric functions, it might be possible in certain cases to get around this by changing variables.
The following system, from [30], is a model of a simple power system with a generator slack bus, lossless lines, and a load with real and reactive powers P, Q, respectively. The state variables are (α, V ), where V ejα is the load voltage phasor, and the bifurcation parameters µ are (P, Q). The equations that determine the system equilibria are:
0 = −4V sin α − P
0 = −4V2+ 4V cos α− Q
The system operates at a nominal solution, given by the values (P0, Q0, α0, V0) =
(0.5, 0.3,−0.1381, 0.9078), and shown in Figure 7.3. As the loads P, Q change, the equilibrium moves around, and can eventually disappear. In this problem, we com- pute “safety margins” for the allowable variations in the loads, that guarantee that a saddle-node bifurcation is not reached.
To handle the trigonometric functions, define x := sin α, y := cos α. The first transversality condition is identically satisfied. If for simplicity we do not consider the second generic transversality condition, the equations we need to solve are:
f1 := x2+ y2− 1 = 0
0 0.5 1 Q 0 1 2 P 0 0.25 0.5 0.75 1 V 0 0.5 Q 0 1 2 P
Figure 7.3: Equilibrium points surface and nominal operating point.
f3 := −4V2+ 4V y− Q = 0
f4 := det J =−16V (x2+ y2− 2V y) = 0
Since we are not interested in the case where the voltage is zero, we factor out the first term −16V in the last equation, obtaining:
f40 := (x2+ y2− 2V y) = 0 We would like, therefore, to minimize the function
J (P, Q) := (P − 0.5)2+ (Q− 0.3)2 subject to the equalities above.
Instead of dealing with the problem as a whole, since we have equality constraints in this case it is easier to eliminate the variables that do not appear in the objective. In other words, we will only care about the constraints we can generate that are in the elimination ideal, i.e., hf1, f2, f3, f40i ∩ R[P, Q]. The only reason we do this
−0.5 0 0.5 1 1.5 −1 −0.5 0 0.5 1 1.5 2 Q P P2+4Q−4=0
Figure 7.4: Curve where saddle-node bifurcations occur, and computed distance from the nominal equilibrium.
theoretical viewpoint.
An automatic way of generating this ideal is using Gr¨obner bases, when we use a lexicographic degree monomial ordering [26, 64]. The elimination ideal has only one polynomial, P2 + 4Q− 4. This corresponds to the curve where saddle-node
bifurcations occur; see Figure 7.4. Therefore, to compute a lower bound on the distance from the nominal equilibrium to the closest saddle-node bifurcation, we can find the maximum γ2 that verifies the condition:
(P− 0.5)2+ (Q− 0.3)2− γ2+ λ(P, Q)(P2+ 4Q− 4) is a sum of squares. In this case, it is sufficient to pick λ(P, Q) constant, and we obtain an optimal value of γ2≈ 0.3735, with λ ≈ −0.2883.
To verify that the restriction to the elimination ideal is not crucial, we can easily verify that multiplying the expressions
by f1, f2, f3 and f40 respectively, and adding, we obtain the valid constraint P2 +
4Q− 4 = 0. Therefore, the only difference in that case would be the need of using nonconstant multipliers.
Though not guaranteed a priori by the method, in this case again we obtain a bound that is exact. As seen in the figure, there exists a solution of the equations that achieves the computed value of γ2, corresponding to P ≈ 0.7025, Q ≈ 0.8766.