The power system models presented in the previous section can be described in compact form as a set of time-invariant nonlinear DAEs:
0=F( ˙χ(t),χ(t),u(t)), (2.25) where the vectorχ∈Rnχ includes the state variables of a power system, e.g. the synchronous generator rotor speedδand the bus voltageV, the vectoru∈Rnu contains to the system inputs, such as controller set-points and disturbances, and 0 is a vector of zeros with proper dimension. Note that the time dependency is often omitted for simplicity of notation. Notice that if the Jacobian matrix corresponding to the time-derivative of the state variables is non-singular, i.e. det (∂F/∂χ˙)6= 0, then, the DAE simplifies to an explicit set of ordinary differential equations (ODEs). In other words, DAEs can be interpreted as a set of ODEs subject to a set of algebraic constraints. Loosely speaking, the degree of complexity to transform (2.25) into an explicit ODE system is determined via the so-called DAE-index; the index refers to the number of differentiation steps required to find a description of the time-derivatives for all state variables. Clearly, as the DAE-index gets higher, the more difficult will it become to solve the set of DAEs numerically, since no analytical solution exists for this class of equations. Throughout this thesis we only consider index-1 DAEs; this is a fairly general assumption that holds for many practical problems, especially for the standard power system models presented in Sec. 2.1.
2.2.1 Objective
The objective of the chapter is to introduce the reader -unfamiliar with reachability analysis- to the computation of reachable sets for the class of index-1 DAE systems. Note that we employ reachability analysis throughout this thesis to
• Chapter 3: Analyze transient stability of power systems in a compositional manner.
• Chapter 4: Estimate the region of attraction of an equilibrium in power systems.
• Chapter 5: Synthesize a set of decentralized linear-parameter varying (LPV) controllers to robustly establish transient stability with formal guarantees of multi-machine power systems.
• Chapter 6: Verify safety of critical components found in power plants.
As mentioned earlier, reachability analysis basically determines the set enclosing all possible trajectories of differential and algebraic variables over a user-defined time-horizon. A definition of reachable sets is given as follows:
Definition 2.1. Reachable Set: Given an implicit DAE system described as in (2.25), the reachable set of differential and algebraic variables over the time-horizon t ∈ [0, tf], where tf is the final time,
starting from the set of consistent initial statesR(0) and the set of uncertain inputs U, is defined as:
reach(R(0),U, tf) :=
χ(t) : χ(t) satisfies (2.25) within [0, tf] for [ ˙χ(0), χ(0), u(t)]∈R(0)×U
.
It is worth noting that Def. 2.1corresponds to the exact reachable set. In fact, except for very specific classes of systems, exact computation of reachable sets is difficult or even impossible [111]; thus, existing techniques aim at introducing traceable and efficient numerical procedures to compute an over-, or under-approximation of the reachable set as illustrated in Fig. 2.7. In this thesis we mainly consider computation of over-approximative reachable sets; that is in other words, an outer-approximation enclosing as tightly as possible all behaviours of the nonlinear DAE system such that:
R([0, tf])⊇reach(R(0),U, tf), (2.26) withR([0, tf]) denoting a superset of the exact reachable set.
Exact reachable set
Inner-approximation of the reachable set
Outer-approximation of the reachable set
Figure 2.7: Projection of the inner- and outer-approximation of an exact reachable.
2.2.2 Existing Techniques
The techniques for reachability computation are generally categorized into two classes of methods [98];
that is either Eulerian schemes based on level set methods (LSMs) or Lagrangian techniques that follow the flow of the system’s underlying dynamics. Shortly after we only consider Lagrangian reachability computation since the algorithm employed throughout this thesis is based on this class of techniques.
Note that Eulerian methods are addressed later in this thesis; specifically in Ch. 4when we consider the estimation of the ROA via the computation of backward reachable sets.
The Lagrangian techniques compute reachable sets similarly to numerical integration methods; that is in other words, by propagating the set of reachable states instead of only computing the solution for a
single point over the specified time-horizon. One consequence of this is that Lagrangian approaches can handle higher-dimensional systems, where the associated memory requirements grow moderately with the system dimension, depending the choice of the reachable set representation discussed shortly after in the following section. A simplified illustration of this concept for a generic system is shown in Fig. 2.8.
There exists a large variety of well-developed methods that considers nonlinear ODE systems, such as abstraction using local linearization [13,56] and Taylor models [29], however, there is little work regarding an efficient algorithmic procedure for the computation of reachable sets for DAE systems that can scale towards industrially-relevant problem sizes. One obvious reason is that an extension of the reachability algorithms based on Lagrangian schemes for ODEs to handle DAEs is necessary; this task, however, is not straightforward since the class of DAE systems differs in both theoretical and numerical properties [37].
Initial reachable set
Final reachable set
Propagated reachable set
Figure 2.8: Illustration of the propagation of the reachable set spanned through the time-horizont∈[0, tf].
So far all the results reported in the literature tackle DAEs using two approaches in order to exploit the efficient methods developed for the class of ODEs: (1) They perform a local linearization around a stable equilibrium point in order to bring the system to a set of explicit ODEs via its index-1 property.
(2) Alternatively, they make several assumptions to eliminate the set of algebraic equations inherently present in the system formulation; hence bringing the system to a set of ODEs. Note that a detailed literature review about existing works that employ reachability computation for the analysis of power system is provided later in Ch. 3.
In this thesis, we employ the first approach performing a local linearization. The reason behind this choice is that the assumptions employed in the literature to eliminate the set of algebraic constraints are often unrealistic and do not meet practical requirements. These assumptions are addressed later in this chapter when we illustrate the applicability of reachability analysis to several benchmark examples.
The remainder of this introductory chapter is organized as follows: First we introduce some basics about set representation in Sec. 2.3, then in Sec. 2.4 we recapitulate from [6, 12] well-know techniques for computing over-approximative reachable sets for the class of DAE systems. Finally in Sec. 2.5we apply
the reachability algorithms discussed throughout this chapter on some examples, with a particular focus on studies involving transient stability of power systems.