Equal Area Criterion

+K₃ (4.16)

Usingδ_s0^pre the unstable equilibrium point (u.e.p.) δ_u0^pre can be found as δ_u0^pre =π+K₃−arcsin

P_ref −K_1p^pre K₂^pre

=π−δ_s0^pre+ 2K₃ (4.17) From the conclusions related to theK’s it is evident that the pre- and post-fault states are identical when using the classical model, and thus the equilibrium points of the two states are also identical as long as the power referenceP_ref is constant, i.e.

δ^pre_s0 =δ^post_s0 δ^pre_u0 =δ^post_u0 4.2.2.2 Using the AVR Model

When using the AVR model the pre-fault system will be identical to the pre-fault system when using the classical model and thus the pre-fault e.p.’s are still calculated using (4.16) and (4.17).

However, as E is now changing, the K’s need to be continuously updated withE given by the dynamics both during fault and post-fault, and the known pre-fault value ofEis therefore denoted E^pre. This also implies that the e.p.’s of the post fault system will depend on E as given by the dynamics. Lastly, sinceE is a known quantity in the pre-fault state andω=ω_nin steady-state, the initial value ofM_fi^pre_f is found as

Mfi^pre_f = E^pre

ω (4.18)

4.3 Equal Area Criterion

With the system dynamics modelled analytically, it is possible to start the stability investigation.

Such an investigation typically seeks to determine two parameters that is of particular interest, namely the Critical Clearing Angle (CCA),δ_cc, and the corresponding Critical Clearing Time

(CCT)t_cc. The critical clearing time is the maximum permissible duration a fault can last for which the system will remain stable after the fault. As such it is the time it takes from the initialisation of the fault untilδreaches the critical angleδ_cc, after which the system will not be able to regain steady-state. To determineδ_cc and the correspondingt_cc, mainly two methods are used in the classical stability assessment, and the first method that will be adopted to the VSM is known as the Equal Area Criterion (EAC).

The EAC is commonly adopted in transient stability analysis of SG’s by using the classical model without dampening, and the dynamics in (4.9) will therefore be considered in this section.

4.3.1 Introduction to the EAC

The Equal Area Criterion can be explained intuitively using theP −δcurve depicted in Figure 4.3, where the pre- and post-fault states are identical as per the problem formulation and the chosen dynamics. A more in-depth explanation can be found in [58].

During pre-fault the system is steady, operating atδ_s0 in point1. A fault then lowers the grid voltage, and the operating point moves from point1on curve I to point2on curve II. By recalling the analysis in Section 2.8.2, asP_ref > P_ethe virtual rotor will accelerate according to the swing equation. The operating point will then move along curve II from point2to point3where the fault is cleared atδ =δ_c. The operating point then jumps to point4on curve I, whereP_ref < P_e. Again, recalling the analysis in Section 2.8.2, the rotor angle will decelerate until point5where

∆ω = 0andδ =δ_max, before accelerating back towardsδ_s0in point1. The angleδ_maxis thus the maximum angle of the post-fault system, and it is known thatδ_max < δ_u0 yields a stable post-fault system asδmoves back towards the s.e.p. instead of diverging.

Figure 4.3: TypicalP −δcurve, modified from [56].

As will be shown, the mathematical derivation of the EAC involves integration of powers. As such, it is evident that the EAC uses energy as an instrument to determine for which clearing

angles δ_c the system will remain stable. Using the discussion above we define the areas A_a and A_d in Figure 4.3 as the acceleration area and the deceleration area respectively. A_a will then represent the injected kinetic energy during fault, whileA_dwill represent the ability of the system to absorb energy. For the system to remain stable, it should be able to absorb all the power that is being injected during fault. This can be used to define the EAC; the system will remain stable as long as the deceleration area is equal to, or larger, than the acceleration area, i.e A_d ≥A_a.

The mathematical derivation used to perform a stability analysis using the above-explained concept of the Equal Area Criterion is provided in Section 4.3.2.

4.3.2 Mathematical Derivation

From the introduction to the EAC it is evident that a condition for the system to remain stable is that at at some time t_a after the fault has been cleared the system reaches a state where

dδ

dt = ∆ω(ta) = 0 before reaching the unstable angle. This can be used to derive the EAC mathematically, and the following derivations will largely follow the discussion in [34].

With the dynamics given by the classical model without dampening from (4.9), the swing equation is given as

It is now possible to multiply both sides with ^dδ_dt = ∆ωyielding

∆ωd∆ω

Next, by integrating this expression,∆ω(t_a)can be found in (4.19) through the following steps:

Z ∆ω(ta)

Recalling that ^dδ_dt = ∆ω(t_a) = 0must be satisfied for the system to remain stable yields equation (4.20). The system can then be classified as stable if there exists an angleδ=δ_max so that the condition in (4.20) holds[34].

Equation 4.20 can be used to define the previously explained acceleration- and deceleration areas.

This is done by definingδ=δ_cas the clearing angle of the VSM when clearing the fault at the clearing timet_c. Equation (4.20) is then reformulated as in (4.21) using the fault- and post-fault quantities, whereA_arepresents the acceleration area andA_drepresents the deceleration area.

Z δc Criterion. From Section 4.3.1 and Figure 4.3 it is clear that the maximum deceleration area available is obtained when the maximum angle is the unstable e.p. of the post-fault system, i.e. δ_max = δ_u0^post. The critical clearing angleδ_cc can then be found as the angle fulfilling the condition in (4.22) withδ_max =δ_u0^post. The critical clearing angle fulfilling the above condition is then found by solving f(δ_cc) = A_a−A_d,max = 0, typically by using software such as MATLAB. The expressions in (4.23a) and (4.23b) are therefore derived from (4.22) to simplify software implementation.

A_a = (P_ref −K_1p^f )(δ_cc−δ^pre_s0 ) +K₂^fh Withδ_ccfound, it is of interest to find the correspondingt_cc. However, using the EAC, this will only be possible if the electrical power is constant during the fault, i.e. P_e^f =constant. This will only be the case ifK₂^f = 0. OtherwiseP_e^f will vary withδwhich is changing according to the dynamics. AsEis considered constant in the classical model, the only scenario yielding K₂^f = 0is a fault whereV_g^f = 0. If this is the case, the electrical power during fault is given by K_1p^f and the critical clearing timet_cccan be found analytically using (4.24) which is derived by using the the swing equation in the following way:

d∆ω

⇒δ_cc−δ_s0^pre = Sb

2ω_nJ

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.