\section 7. FAST AND SLOW TIME-SCALES %[last updated January 22] \subsection 7.1 Time-scale decomposition The fast-slow time scale decomposition is carried out using the analysis known as singular perturbations [\Kokotovic,\OMalley]. The standard model of a two time scale system is: $$\eqalign{ \dot{x}&=f(x,y)\cr \epsilon \dot{y}&=g(x,y)}$$ where $x$ is the ``slow'' state vector, $y$ is the ``fast'' state vector and $\epsilon$ is a small number. The first approximation to a time scale decomposition is the assumption $\epsilon\rightarrow 0$, in which case the second equation becomes algebraic corresponding to equilibrium conditions for the fast variables. Therefore, the slow component $y_s$ of the fast state variables $y$ can be evaluated as function of the slow variables $x_s$. Thus the approximate {\sl slow} subsystem is defined by the following differential-algebraic equations: $$\eqalign{ \dot{x_s}&=f(x_s,y_s)\cr 0&=g(x_s,y_s)}$$ This is the quasi steady-state representation of a two time scale system. Further approximation is possible using an expansion in powers of $\epsilon$, but this is beyond the scope of this brief presentation. We illustrate the quasi steady state approximation in Figure 2.7-1 in the case of a two state system with one fast variable $y$ and one slow variable $x$. The equilibrium condition $g=0$ defines a curve in the $xy$ plane, which we call the fast dynamics equilibrium manifold (in this two dimensional system the curve is called a manifold so that the terminology applies to multivariable systems as well). When $\epsilon$ is very small, this is a good approximation of the ``slow manifold'' of the two time scale system. The equilibria of the full system are the points on the manifold defined by $g=0$, for which also $f=0$. In Figure 2.7-1 two such equilibria are shown, one stable and one unstable. Each point $x_s$, $y_s$ of the fast dynamics equilibrium manifold is the equilibrium point of a fast subsystem defined as: $$\epsilon\dot{y}_f=g(x_s,y_s+y_f)\eqno(\hbox{2.7-1})$$ where $y_f=y-y_s$ is the fast component of $y$. The time scale decomposition is valid only when the fast subsystem defined above is stable at its equilibrium point $y_f=0$. With this assumption, the behavior of the two time scale system can be approximated as follows: For an initial condition outside the fast dynamics equilibrium manifold a fast transient is excited at first. One common possibility is that the fast transient acts to put the system state onto the fast dynamics equilibrium manifold before the slow variables have time to change considerably. For example, an initial condition such as point A on Figure 2.7-1 leads to a fast downwards transient to the upper portion of the fast dynamics equilibrium manifold. Following this fast transient, the system will remain on the fast dynamics equilibrium manifold and it will slowly move towards the stable equilibrium. When large disturbances are considered, the existence of a stable fast dynamics equilibrium after a disturbance is not the only requirement for a valid time scale decomposition: The pre-disturbance state of the system must also belong to the region of attraction of the post disturbance stable equilibrium of the fast dynamics. For the system of Figure 2.7-1 the region of attraction of the stable part of the fast dynamics equilibrium manifold is easily determined: all initial conditions above the unstable part of the fast dynamics equilibrium manifold are attracted to the stable part. On the other hand, an initial condition lying below the unstable part of the fast dynamics equilibrium manifold initiates a collapse, even though a stable equilibrium still exists. % omitted by ian since too specialized %When the fast dynamics lose stability through a Hopf bifurcation, the region %of attraction of a stable equilibrium point near the bifurcation depends on %whether there exists an unstable limit cycle associated with the HB. When the %HB is subcritical, the boundary of the region of attraction of the stable points %near the bifurcation includes the stable manifold of the unstable limit cycle. \subsection 7.2 Saddle node bifurcation of fast dynamics As the slow dynamics drive the system along the fast dynamics equilibrium manifold, the fast subsystem defined above changes and the fast dynamics may lose stability. If the slow dynamics are thought of as slowly varying parameters, then the instability of the fast dynamics may be understood as a bifurcation of the fast dynamics [\Cutsemninesix]. In the fast equations (2.7-1), $x_s$ may be thought of as the bifurcation parameter (note that $y_s$ depends on $x_s$). (We often expect the slow dynamics to arise from the disappearance of the operating equilibrium due to a disturbance. In this case it should be noted that stability is already lost before the bifurcation of the fast dynamics in which the fast dynamics lose stability.) % this is perhaps too technical; the same problem occurs for section 2.4 % when the saddle node bifurcation dynamics are slow and comparable to the parameter variations %As stated above, when the fast dynamics are unstable, the time scale decomposition %is no longer valid. However, an approximate analysis is still possible. Consider, for instance, a system for which the fast dynamics equilibrium manifold is the nose curve of Figure 2.7-2. Point B is a saddle node bifurcation of the fast dynamics. The fast subsystem is stable on the upper part of the fast dynamics equilibrium manifold and unstable on the lower part of the fast dynamics equilibrium manifold. If $\epsilon$ is assumed sufficiently small, the fast dynamics are approximated by vertical lines moving towards stable points of the fast dynamics equilibrium manifold and away from unstable points of the fast dynamics equilibrium manifold. In this particular system, the slow dynamics are such that the slow state $x$ always increases. Consider now the response of the system starting from an initial point A lying above the nose curve. At first the fast dynamics will drive the system to the stable upper part of the fast dynamics equilibrium manifold. This will be a fast transient. Then the system will move slowly along the fast dynamics equilibrium manifold driven by the slow dynamics. This process can continue until point B is reached. At B, the two fast dynamics equilibria coalesce in a saddle node bifurcation. The dynamic consequence of the bifurcation is collapse of the fast dynamics as the state follows the vertical arrows near B. % omitted by ian since too specialized %It should be noted that the system trajectory will also depart from the %curve $g=0$ after a Hopf bifurcation of fast dynamics, i.e. when the linearized model of the %fast subsystem has a pair of purely imaginary eigenvalues. At such a point the %system will depart from the fast dynamics equilibrium manifold with oscillations of increasing amplitude. \insertfigure \insertfigure \insertfigure \insertfigure \subsection 7.3 A typical collapse with large disturbances and two time scales Let us illustrate a typical collapse triggered by large disturbances and involving fast and slow dynamics. The system is initially at a stable equilibrium and the following sequence of events takes place: \item{(1)} A disturbance happens and the system restabilizes \item{(2)} A second disturbance happens and the operating equilibrium is lost. (This is the large disturbance equivalent of a saddle node bifurcation as discussed in Section 2.4.) \item{(3)} Due to this loss of equilibrium a slow collapse begins and lasts for some time \item{(4)} In this case, the slow collapse leads to a saddle node bifurcation of the fast dynamics which causes a faster collapse and hence a total system disruption. \noindent (This chapter defines the collapse to begin with the instability (2) and to include the slow and fast dynamics of (3) and (4). Some authors prefer to identify the collapse with the fast dynamics of (4) only.) The sequence of events can be illustrated with pictures of the functions $f$ and $g$ in Figure 2.7-3. The two large disturbances are represented by discrete changes in the system equations so that $g$ becomes $g_0$, $g_1$, $g_2$. For simplicity we suppose that $f$ is not affected by the disturbances so that the curve $f=0$ remains the same throughout. The equilibrium points of the various system equations are the intersection points of the fast dynamics equilibrium manifolds $g_0=0$, $g_1=0$, $g_2=0$ with $f=0$. The initial stable equilibrium $S_0$ is the upper intersection of $g_0=0$ with $f=0$. The first disturbance changes $g_0$ to $g_1$ and the resulting transient indicated in Figure 2.7-3 first quickly moves the state to the fast dynamics equilibrium manifold $g_1=0$ and then slowly restores the state to the new stable equilibrium $S_1$. Enough time is assumed to pass so that the restabilization at $S_1$ is achieved. Note that the first large disturbance has reduced the margin to voltage collapse since the system is now closer to a saddle node bifurcation. The second large disturbance changes $g_1$ to $g_2$. A fast transient quickly moves the state to the fast dynamics equilibrium manifold $g_2=0$. Since $g_2=0$ has no equilibrium points, slow dynamics will move the state along $g_2=0$. In Figure 2.7-3, the state moves along $g_2=0$ to the right. The system state will eventually reach a saddle node bifurcation of the fast dynamics and it will depart from the fast dynamics equilibrium manifold $g_2=0$ with a fast transient which is a fast collapse. The second large disturbance changing $g_1$ to $g_2$ is a quick change from a system with two equilibria to a system with no equilibria. If the large disturbance were instead thought of as a gradual change, the system would pass through a saddle node bifurcation at which the equilibria coalesced and disappeared as described in section 2.4. Now we give a more concrete example of the more general collapse above by choosing to think of the fast dynamics as the network transients and the slow dynamics as the load recovery to constant power. (For simplicity, the load is assumed to be constant power in steady state.) In terms of the variables of the load model of section 2.3, $y$ is identified as the load voltage $V$ and $x$ is identified as the internal load state $x_p$. Then the curves $g_0=0$, $g_1=0$, $g_2=0$ represent the network capability and the large disturbances could be caused by network outages. The curve $f=0$ represents the constant load power in steady state. ($f(x_p,V)$ could be the right hand side of equation (2.5-1).) In Figure 2.7-3, the system is presented with the slow variable $x_p$ on the horizontal axis. Since $x_p$ is the slowly varying variable, the saddle node bifurcation of the fast dynamics occurs at the nose of the fast dynamics equilibrium manifold $g_2=0$. It is often useful to present the system with instantaneous real power $P$ on the horizontal axis. This skews the diagram so that it appears as in Figure 2.7-4. In Figure 2.7-4, the fast dynamics move at angle so that typically both voltage and power drop quickly when a disturbance occurs (c.f. section 2.5). Also the constant power characteristic $f=0$ appears as a vertical line. Figures 2.7-3 and 2.7-4 present two views of the same collapse and it is useful to understand both views when reading the literature.