Single-Machine Infinite Bus

ω_j = 1

2Hj

(P_m,j−P_e,j−D_j(ω_j−ω_ref)),

with: Pe,j =E⁰_q,jVjY_j^gcos(Θ^g_j+δj−θj)−V_j²Y_j^gcos(Θ^g_j),

(2.70)

and the power-flow equations:

Pi=Pe,i−Pl,i=Vi

X

k∈N

ViYikcos(Θik−θk−θi), Qi=Qe,i−Ql,i,=−Vi

X

k∈N

ViYiksin(Θik−θk−θi).

(2.71)

Here the subscripts j andi specify the j-th machine and the i-th bus, respectively. It is assumed that the mechanical power P_m and the generator voltage E_q⁰ are kept constant. The constants Y^g and Θ^g denote to the absolute value and the phase angle of the admittance spanned from the generator to its corresponding bus. All remaining variables and parameters were previously introduced as in Table 2.1 and Sec. 2.1.2.

Remark 2.13. Notice that the DAE system described via (2.70) and (2.71) can be simplified to an explicit ODE system if we replace the bus voltageV_j by a constant value which in turn eliminates the need to solve the set of algebraic equations associated with the power-flow equations (2.71). Clearly this assumption is unrealistic and does not hold for many practical situations. Note that this is a basic, and necessary, assumption employed to establish transient stability via Lyapunov direct methods [118]. This assumption is not required to assess transient stability using reachability analysis.

Remark 2.14. We simplified the models in (2.70) just to focus on the demonstration of the reachability algorithm for readers unfamiliar with reachability analysis. It should be stressed that we employ the introduced power system models without any modelling simplification in the following chapters.

2.5.1 Single-Machine Infinite Bus

The first example is the so-called SMIB system. The system illustrated in Fig. 2.13 consists of a synchronous generator connected to an infinite bus; a special bus whose voltage and phase angle are kept constant regardless of the changes occurring in the network. The SMIB power system is compromised of six state variables: two variables corresponding to the generator dynamic states appearing in (2.70) and four algebraic variables associated with the constraints at the generator bus.

2.5.1.1 SMIB Linear Model

Here we shall consider the linear model of the SMIB system.

Example 2.2. The linearization of the SMIB power system at its equilibrium point results in the LDI

with the state variablesx= (δ, ω)^T, andU as a set of input disturbance affecting the frequencyω.

The reachable set is computed according to Alg. 1 starting from the initial reachable setR(0), over a time-horizon tf = 5 s under the influence of the set of uncertain inputs U. The chosen time-increment is t_r = 0.005 s and the number of Taylor terms specified for the matrix exponential is σ = 6. The time-domain projection of the reachable set is shown in Fig. 2.14 in addition to randomly generated trajectories (n= 20). Notice how the reachable set encloses all possible system trajectories by running the reachability algorithm just once, whereas one needs to run several simulations (infinitely many) in order to approximate the same result.

Transmission line 1

Figure 2.13: The single-machine infinite bus benchmark problem.

2.5.1.2 SMIB DAE Model

Now we consider the SMIB system modelled via the set of nonlinear DAEs described in (2.70) and (2.71).

Example 2.3. Here the fault scenario under consideration is the loss of the second transmission line att= 0.01 s, followed by its reconnection to the network after clearance of the faultt= 0.02. The reach-able set is computed according to Alg. 3 starting from R^x(0) = ([0.65075,0.66675] [−0.008,0.008])^T. We include uncertainty in the initial set of differential variables, since initial states are not exactly known due to increasingly varying operating conditions in current power systems. Here we do not specify the terminal time tf, instead, the reachability algorithm runs until all states are enclosed by R(0), which occurs att= 0.23.

The chosen time-increment is tr = 0.0006s and the number of Taylor terms specified for the matrix exponential is σ = 6. Fig. 2.15 shows projection corresponding to the reachable set of the differential

Over-approximative reachable set

(gray area) Random trajectories

(solid lines)

Initial reachable set (red box)

Terminal reachable set (green box)

Figure 2.14: Time-domain bounds of the reachable set of Example2.2.

state variablesδandωstarting from the initial reachable setR^x(0). Fig. 2.16illustrates the time-domain bounds of the algebraic constraints y = (P, Q, V, θ)^T at bus 1. Here the discontinuous jump occurring in the reachable set is associated with the fault scenario. Immediately after losing the transmission line, entries of the admittance matrix (2.5) change thus leading to a discontinuous jump in the algebraic variables to satisfy the power flow equations (2.71).

2.5.1.3 Effect of the Algorithm Parameters

Now we discuss the effect of the parameters associated with the reachability algorithm; namely we consider the time-incrementtrand how the set of the Lagrangian remainders is computed.

Example 2.4. In Fig. 2.17(a), the set of linearization errors is computed according to the conservative method in (2.69), see Remark (2.11), rather than the tight technique using the quadratic mapping of zonotopes suggested in Prop. 2.3. It can be seen that after four time steps, the constraining condition L⊆Lmaxwas not fulfilled, and Alg. 3terminates as the reachable set is no longer converging to the initial operating point. This is clearly not true, as we just demonstrated that the power system is clearly stable for the considered fault scenario. Here, the reachability algorithm fails to converge as the computation of the setLled to unacceptable over-approximation of the reachable set.

Reachable set of the fault trajectory (dark-gray area)

Reachable set of the post-fault trajectory (gray area)

Initial reachable set

Figure 2.15: Reachable set projection of the differential variablesδandωof the SMIB power system. The dark-gray area show the reachable set during the fault, and the gray area specifies the reachable set of the post-fault trajectory. The considered fault scenario is the loss of the second transmission line connecting the synchronous generator to the infinite bus. The line is reconnected after the clearance of the fault, and the reachable set is computed until all states are enclosed by the initial set of statesR(0).

Discontinuity to satisfy the algebraic power-flow equations

Figure 2.16: Time-domain bounds of the algebraic constraints of the SMIB power system. Here the variablesP, Q, V, θchanges instantly att= 0.01 andt= 0.02 in order to satisfy the power flow equations (2.71). This leads to the discontinuity of the reachable set.

Example 2.5. In Fig. 2.17(b), we increasedtr= 0.0006stotr= 0.006s. Notice howtraffects the over-approximation of the particular solution, according to Prop. 2.2, associated with the LDI (2.47). Clearly, the larger the time-increment, the more conservative the particular solution becomes and convergence of the reachable of the solution is no longer guaranteed as seen after several time-steps in Fig. 2.17(b).

Remark 2.15. The outcomes of reachability analysis using the CORA toolbox heavily relies on the chosen parameters for the analysis. Improper choice of parameters can result in an unacceptable over-approximation although reasonable results could be achieved by using appropriate parameters as illus-trated in the previous examples. Currently a self-tuning algorithm which specifies the optimal parameters for reachability analysis is being investigated as part of future work at the institute.

(a) (b)

Figure 2.17: Influence of the algorithm parameters on computation of the reachable sets. In (a) the set of linearization errors is computed according to the conservative method proposed in (2.69). In (b) the time-incrementtris slightly increased resulting in unacceptable over-approximation of the reachable set.