Virtual Damper Windings

The original Synchronverter from [27][28] combines the governor droop function with the damping, purely based on the VSM’s speed deviation from the nominal frequency. However, as pointed out in [68], this may lead to poorly damped dynamics when the frequency droop rate is chosen sufficiently large, i.e. D_pis tuned to be small. Therefore, [68] lists several other possible implementations of damping to the VSM found in the literature, including the use of frequency measurement by use of PLL, using the frequency derivative, and even implementing a power forward term. In addition, a simple, yet challenging, implementation of a Power System Stabiliser (PSS) to dampen the voltage amplitude is proposed. However, all of the proposed damping methods have the inherent property of requiring tuning which in many cases can be both challenging and time-consuming.

Therefore, to further improve the controller performance when subjected to a large contingency, a novel implementation of virtual damper windings to the Synchronverter VSM will be invest-igated. Damper windings are well-known to reduce mechanical oscillations and hunting in the conventional Synchronous generator[44] without the need of additional tuning, and the method proposed here will add the damping power to the power set-points of the Synchronverter. The derivation of the electrical equations revolving around the damping power in the conventional SG are long and tedious. Therefore, only the essential parts needed to derive the control method are included here. The explanation of the damper windings will largely follow the derivations in [34], and is based on the same assumptions as was used in Section 2.3, except that damper windings are now included.

It is known that the stator of a Synchronous generator is represented by the three magnetic axes a,bandc, corresponding to the phase windings, while the rotor is represented by the pole axis (d-axis) and the inter-pole axis (q-axis)[34]. The rotor will be equipped with two short-circuited

damper windings, one in thed-axis and one in theq-axisas depicted in Figure 5.3. In Figure 5.3,β_m is the angle of the rotor with respect to the stationary referencea-axis, and equals the angleθof the of the Synchronverter. Furthermore,L_aa,L_bb, andL_ccare the self-inductances of the stator windings, whileL_{f f},L_DD, andL_QQare the self-inductances of the field- and damper windings respectively.r_a,r_b,r_c,r_f,r_D andr_Q are the related resistance of each of the windings.

Figure 5.3: Structure of a synchronous generator including damper windings[34].

Since the windings are magnetically coupled, the flux linkages are given by



whereL_mn = L_nm is the mutual inductance between windingm andn. It can be shown that when neglecting the two damper windings, the linkages in (5.8) are identical to the flux linkages derived in (2.14). Since many of the inductance elements vary with the angle, it is common to transform the flux linkages from the stator reference frame to the rotor reference frame, i.e. from abc-components todq0-components which rotate together with the rotor. This can be done by using the well-known Park’s coordinate transformation in (5.9) and (5.10).

P =

v_dq0 =P v_abc i_dq0 =P i_abc Φ_dq0 =PΦ_abc

(5.10)

Using Park’s coordinate transformation and applying mathematical derivation, the flux linkages in (5.8) are rewritten into three magnetically decoupled winding sets as in (5.11)[34].



Here the notation is changed so thatM represents the mutual inductance andk = q3

2. Recalling that for balanced three-phase systems i₀ = 0yieldsΦ₀ = 0. It is now possible to extract the parts of (5.11) that are of interest for the introduction of virtual damper windings to the VSM.

Only the flux linkages of thed- andq-axis are of interest, i.e. Φ_dandΦ_q, which can be found from (5.11) as

Φ_d=L_di_d+kM_fi_f +kM_Di_D (5.12a)

Φq =Lqiq+kMQiQ (5.12b)

Using (5.12a) and (5.12b), the PCC voltage, i.e. the terminal voltage of the VSM, in dq coordinates can be expressed as in (5.13a) and (5.13b) where R_sis the resistance of the filter inductanceL_s,L_d=L_q =L_sfor the VSM andωis the converter speed.

v_d =−ωΦ_q−R_si_d=−ωL_qi_q−ωkM_Qi_Q−R_si_d (5.13a) v_q=ωΦ_d−R_si_q =ωL_di_d+ωkM_fi_f +ωkM_Di_D−R_si_q (5.13b) Equations (5.13a) and (5.13b) can now be used to find expressions for calculatingM_Di_D and M_Qi_Qwhich are the flux contributions from the damper windings onto thed- andqvoltages.

MDiD = v_q−ωL_di_d+R_si_q−ωkM_fi_f

ωk (5.14a)

M_Qi_Q= v_d+ωL_qi_q+R_si_d

−ωk (5.14b)

Thus, utilising the already available measurements of v_pcc and i_c,grid and applying the Park transformation, bothM_Di_D andM_Qi_Q can be calculated using (5.14a) and (5.14b). BothM_Di_D

andM_Qi_Qare zero in steady-state and will therefore not change the steady-state characteristic of the converter. As soon as a transient occurs,M_Di_D andM_Qi_Qbecomes non-zero and can thus be used to implement damping power into the set-points of the VSM. The expression forM_Di_D will however need some slight adjustments to serve the desired purpose for the applications in this thesis. In its current form,M_Di_D will return to zero even before the fault is cleared if the fault clearing time is long, yielding a good response to small-signal disturbances but limited effect during large, longer-lasting disturbances. Moreover, it is desired to have the effect of the damper windings also after the fault has been cleared, until the system has again reached its nominal operating points.

In (5.14a), the expression for E is recognised as the last part. Looking more closely it can be deduced that as long asv_q plus the voltage drop across the filter resistance and inductance equals the back-EMF of the VSM,M_Di_D will be zero. As described above, we want to have a contribution from the virtual damper windings as long as the VSM does not operate at its nominal set-points. Therefore, the actual back-EMF,E =ωM_fi_f, is replaced with the nominal back-EMF E_n = ω_nM_fi_f,n in the expression for M_Di_D yielding (5.15), where both the measured PCC voltage and the measured converter current have been transformed using Park’s transformation.

In theory, this will yield a contribution from the artificial damper windings, i.e. M_Di_D 6= 0, for as long as E 6= E_n, due to the fact that v_q plus the voltage drop across the filter inductance, and related resistance, will not be equal toE_n. This is beneficial considering the dynamic ofE when operating far from the equilibrium point immediately after the fault has been cleared. The modifications will not impact the steady-state characteristic of the VSM asM_Di_D will still be zero in steady-state whenE =E_n.

M_Di_D = v_q−ωL_di_d+R_si_q−ω_nkM_fi_f,n

ωk (5.15)

Having calculated both M_Di_D and M_Qi_Q using (5.15) and (5.14b) respectively, the damper windings contribution to the VSM set-points can be calculated using (5.16a), (5.16b) and (5.16c), whereiis the measured three-phase converter current andcosf θandsinfθare still defined as in (2.12). Again,h·,·idenotes the inner product inR³.

T_D =−M_Di_Dhi,sinθf i+M_Qi_Qhi,cosθf i (5.16a) P_D =−ωM_Di_Dhi,sinθf i+ωM_Qi_Qhi,cosθf i (5.16b) Q_D =ωM_Di_Dhi,cosθf i −ωM_Qi_Qhi,sinθf i (5.16c) T_D andQ_D as calculated by (5.16a) and (5.16c) respectively will both be positive forE < E_n. The control signals are then added to the APL and RPL respectively as depicted in Figure 5.4.

The virtual damper windings are here added to the system already equipped with the virtual resistor to further improve the dynamic response. It is however also possible to implement the virtual damper windings to the original Synchronverter without adding the virtual resistor, as the two implementations are entirely independent of each other. It should be noted that for the actual controller implementation, the damping power is added before the set-point saturation described in Section 3.1.5 to ensure that the set-points, including the contribution from the artificial damper windings, are within the operating capability of the converter.

Calculation of

Switch A

Amplitude detection Switch B

PWM generation

2 1 MPPT

Active if

Figure 5.4: Enhanced control structure including virtual resistor and damper windings.

The virtual damper windings proposed in this section use the back-EMF of the VSM to implement damping, without the need for additional tuning. The mathematical model is modified to improve the dynamics under both small-signal disturbances causing oscillations, and improve stability when subjected to a large contingency. The method is however designed based on the assumption of the nominal voltage also being the reference voltage. If the converter is set to operate away from the nominal voltage, the enhanced control structure would introduce a steady-state deviation, and a subsequent need to be redesigned.

A similar method as the one described above was proposed in [69] for a High Voltage Direct Current (HVDC) system. However, the implementation into the control algorithm proposed here is vastly different and modified to improve damping during longer and more severe contingencies while still achieving the same advantages, all with less controller complexity. As such, when combined with the virtual resistor from Section 5.2.2, the implementation of both a virtual resistor and virtual damper windings comprise a novel control strategy, having the potential to drastically improve the dynamic performance of the Synchronverter control topology when subjected to a severe contingency.

5.3 Simulations and Results

Simulations have been carried out to validate the effectiveness and functioning of the enhanced control loops described in Chapter 5. In addition, for the system added with the power correction loop, the quasi-steady approximate Lyapunov approach is modified to include the changed fault dynamics to check whether the method can still provide adequate results for the stability limits when the loop is added to the system. The system parameters used in the simulation model are the same as for the original system. When discussing the performance of the enhanced systems, the two main objectives of the added control loops should be kept in mind; improve the transient stability limits of the system, and mitigate the excessively high post-fault converter current.