TEF Without Damping

P_ref −P_e^f

t²_cc

t_cc =

s 2ω_nJ

S_b(P_ref −Pe^f)(δ_cc−δ^pre_s0 ) (4.24) For situations where K₂^f 6= 0the critical clearing time cannot be found analytically using the EAC, and other methods must therefore be applied. Thus the Equal Area Criterion has been adopted to the Synchronverter VSM to find the critical clearing angle and, if possible, the critical clearing time. This concludes the stability analysis using the EAC.

4.4 Transient Energy Function - The Lyapunov Function

The second method that will be adopted to the VSM is Lyapunov’s direct method, i.e. using a Lyapunov function to determine the stability of the dynamical system. The Lyapunov function that is commonly used in power system analysis is also known as the Transient Energy Function (TEF)[32]. This is because an energy function is used as a candidate Lyapunov function, and a thorough study and explanation of the method is given in [59].

The TEF is widely used for assessing power system transient stability due to its speed and reduced processing requirements by computer software. This is because stability can be investigated without the need to solve the system differential equations of the post-fault system, and in contrast to the Equal Area Criterion, the TEF method can also be used to find the CCT in cases where the electric power during fault is not constant. Still, it can be shown that the TEF and EAC are, for all intentional purposes, investigating the same thing, namely the kinetic- and potential energy of the system. This will however not be done here, and interested readers are therefore referred to [32].

4.4.1 TEF Without Damping

First, the TEF will be derived for the Synchronverter VSM using the classical model without dampening, and the derivations will to a large extent follow the discussion in [32]. With the dynamics given by (4.9), the swing equation is given as

d∆ω dt = S_b

ω_nJ

Pref −Pe(δ))

which during normal operation has one stable and one unstable equilibrium point;(δ_s0; ∆ω= 0) and(δ_u0; ∆ω = 0). By rearranging and multiplying both sides with ^dδ_dt = ∆ωwe obtain

ω_nJ Sb

∆ωd∆ω dt −

P_ref −P_e(δ))dδ dt = 0

The TEF will investigate the stability of the post-fault system, and substituting forPeyields ω_nJ

S_b ∆ωd∆ω dt −

P_ref −K_1p^post−K₂^postsin(δ−K₃)dδ

dt = 0 (4.25)

Integrating (4.25) from the stable post-fault e.p. to any point(δ; ∆ω)on the system trajectory during fault results in the following integral equation, whereW_kis the system kinetic energy, andW_p is the system potential energy with respect to the s.e.p..

V = As the left side of (4.25) is equal to zero, the integral of the left side must be constant, meaning the sum of kinetic and potential energy in the system is constant when the damping has been neglected. It is now possible to evaluate the integrals to find the candidate Lyapunov function in equation (4.26).

The derived candidate function must be shown to be a true Lyapunov function satisfying the criteria set out in Section 2.6.3. As the post-fault system is investigated, the e.p.’s of the post-fault system

x₀ =x^post_s0 =

δ_s0^post∆ω₀T

=

δ_s0^post0T

will be used to show that the TEF in (4.26) is a Lyapunov function for the system in question, and thus can be used in the stability analysis.

The first criterion in Section 2.6.3 is easily satisfied by evaluatingV(x^post_s0 ). Substitutingδ =δ_s0^post and ∆ω = 0 into (4.26), it is found thatV(x^post_s0 ) = 0, satisfying criteria 1. The remaining conditions seek to establish that the stationary point for which the candidate function is derived is indeed a minimum point, and not a maximum point or a saddle point as defined in Figure 4.4.

To show that (4.26) satisfies condition2, i.e.

V(x)>0for allx∈ Dexcept atx₀

both the gradient and Hessian matrix must be investigated. First, the gradient must be shown to be zero, and the gradient is therefore calculated using (4.27).

∇V(x) = ∂V(x) By evaluating the gradient at the stable equilibrium point it is shown that the gradient is indeed zero as the electric power and the reference power are equal at the equilibrium, making the s.e.p.

an extrema of the energy function.

Figure 4.4: Different types of stationary points for a scalar function of two variables[32]; (a) minimum, (b) maximum, (c) saddle point

∇V(x)

δ=δ_s0^post,∆ω=0 = 0 0

(4.28) Next, the Hessian matrix,H, is calculated in (4.29), and the objective is to show that the Hessian is positive definite for the stable e.p., which if shown means thatx^post_s0 is a minimum.

H= ∂²V(x)

∂x² =





∂²V(x)

∂δ²

∂²V(x)

∂δ∂∆ω

∂²V(x)

∂∆ω∂δ

∂²V(x)

∂(∆ω)²



=



K₂^postcos(δ−K₃) 0 0 ^ω_S^nJ

b



 (4.29)

The Hessian is found to be a2×2diagonal matrix, and the elements on the diagonal is therefore also found to be the EigenvaluesλofH. For a2×2matrix it can be concluded that the matrix is positive definite if the Eigenvalues are positive[41][60]. The HessianHwill therefore be positive definite when

K₂^postcos(δ−K₃)>0 (4.30a) ω_nJ

S_b >0 (4.30b)

The condition in (4.30a) can be shown to hold for

−π

2 +K₃ < δ < π 2 +K₃

while the condition in (4.30b) is always satisfied. His thus shown to be positive definite on the interval governed byδ above, and as long as δ_s0^post is within this interval, the Hessian will be positive definite at the e.p. x^post_s0 , making the extrema a minimum. The second condition is thus satisfied.

Finally, it must be shown that

V˙(x) = ∂V(x)

∂x ·f(x)≤0for allx∈ D

Substituting using (4.27) and the chosen dynamicsx˙ =f(x)yields h

K_1p^post+K₂^postsin(δ−K3)−Pref ωnJ Sb ∆ωi

·

"

∆ω

Sb

ωnJ

P_ref −K_1p^post−K₂^postsin(δ−K₃)

#

(4.31) and applying some mathematical derivations to (4.31) the third condition is found to be satisfied in (4.32) through he following step:

V˙(x) = (K_1p^post+K₂^postsin(δ−K₃)−P_ref)∆ω+ω_nJ

S_b ∆ω· S_b ω_nJ

P_ref−K_1p^post−K₂^postsin(δ−K₃)

V˙(x) =

K_1p^post+K₂^postsin(δ−K₃)−P_ref

∆ω+

P_ref−K_1p^post−K₂^postsin(δ−K₃)

∆ω= 0 (4.32) The candidate Lyapunov function has thus been shown to be a true Lyapunov function for the s.e.p. x^post_s0 , and can therefore be used for the stability analysis that is to be performed. The idea behind the Transient Energy Function is to check whether the initial conditions of the post-fault state, i.e. the values of the state variables att_c,x_c, lies within the stability region, also known as the attraction region, of the stable post-fault equilibrium point. Such an attraction region is depicted in Figure 4.5, where it can be seen that if the initial condition is inside the region, the system states either oscillates around the s.e.p. if neglecting dampening (Fig. 4.5a) or converges to the s.e.p. if including dampening (Fig. 4.5b), while if the initial condition is outside the region, the system states diverge.

(a) Typical attraction region of undampened system[34].

(b) Typical attraction region of dampened system[34].

Figure 4.5: Typical attraction regions of undampened and dampened systems[34].

Checking whether the initial conditions are inside or outside the attraction region can be done by investigating whether the value ofV(x_c)is less than some critical value indicating the border of the attraction region, i.e

V(x_c)<V^cr

If this is the case, the system will remain stable after the fault. The time whereV(x_c) =Vcr will thus yield the CCT,t_cc. The critical valueV^cr is found by evaluating the TEF in (4.26) at the unstable post-fault e.p.,x^post_u0 . Doing this, the critical value of the Lyapunov function is found using (4.33).

Vcr =K_1p^post(δ^post_u0 −δ^post_s0 ) +K₂^postcos(δ_s0^post−K₃)−K₂^postcos(δ_u0^post−K₃)−P_ref(δ^post_u0 −δ_s0^post) (4.33) The stability analysis is then carried out using numerical integration of the faulted system, using the dynamics in (4.9), whereK₂ is calculated usingV_g^f. For each step in the numerical integration, the TEF is evaluated at the solution of the dynamic system at that time step and then compared to the critical value. The critical clearing time is then found as the time where V(x_tcc) =Vcr, where the critical clearing angle is found asx_1,tcc. This concludes the stability analysis using the TEF based on the classical model when neglecting dampening.

In document Transient Stability Assessment of Virtual Synchronous Machine based Wind Energy Conversion System (sider 78-82)