Generalized Multivariable Grid-Forming Control Design for Power Converters

I. INTRODUCTION

• Grid-forming converters establish frequency and voltage without relying on the grid, synchronize through power balance rather than a PLL, and can support regulation as inverter-interfaced generation grows.Poorly designed grid-forming control can instead deteriorate power-system stability.Source: §I · p. 1
• The surveyed approaches include droop, PSC, VSG, matching control, dVOC, and hybrid angle control [7].They draw on synchronous-generator droop or angle emulation, inertia characteristics, DC-voltage-based frequency, coupled oscillators, or combined AC/DC dynamics.Source: §I · p. 1
• Reported improvements target power-sharing accuracy and damping, add inertia or voltage/frequency coupling, and extend droop, PSC, VSG, and matching control –.Examples include frequency feedback for droop, PI-based virtual inertia for PSC, high-frequency feedback for VSG damping,, and DC-voltage damping for matching control.Source: §I · p. 1
• Earlier work established some equivalences among droop, VSG, and PSC,,,, and presented focused models or comparisons,,.The comprehensive comparison in did not highlight commonalities across controllers.Source: §I · pp. 1–2
• Prior unification studies covered mainly basic architectures, selected controller types, or separate frequency and voltage comparisons, while assuming decoupled SISO loops and relying on burdensome manual tuning.The paper identifies four gaps: improved variations are often excluded, not all controller types are unified, AC/DC and active/reactive loops are assumed decoupled, and nested SISO tuning cannot optimize MIMO performance.Source: §I · pp. 1–2
• The proposed multivariable feedback architecture unifies basic and improved grid-forming controls through a transfer matrix acting on linear and nonlinear error signals.It enables straightforward controller comparison, simultaneous tuning of DC, active-power, and reactive-power loops for performance and robustness, and derivation of new controls.Source: §I · p. 2
• The paper proposes MIMO-GFM control without increasing controller order over basic VSG or droop with LPF, and uses fixed-structure H∞ synthesis to optimize its parameters.The controller is transformed to the standard H∞ synthesis form; the paper addresses all three control loops together.Source: §I · p. 2

II. GENERAL CONFIGURATION OF GRID-FORMING CONVERTER

• The converter model includes DC-capacitor dynamics, representing the DC source as a controlled current source iu in parallel with Cdc rather than assuming ideal AC/DC decoupling.The studied topology connects a three-phase inverter to the grid through a filter; including capacitor dynamics retains information that can improve performance and supports controls based on AC/DC coupling.Source: §II · p. 2

A. MIMO Formulation of Grid-Forming Converter

• The DC dynamics contain AC-side information through Euid, enabling DC signals to support AC-performance improvement; assuming an ideal DC source is not beneficial for optimization.Source: A. MIMO Formulation of Grid-Forming… · p. 3
• The open-loop converter model has three control inputs and five regulated outputs, with grid voltage and frequency treated as disturbances.The MIMO equivalent circuit is shown in Fig. 2.Source: A. MIMO Formulation of Grid-Forming… · p. 3, Figure 2 · p. 3
• The control objective is to choose input u so the system output y meets desired specifications despite disturbance d.The model is abstracted as an open-loop state-space system before multivariable feedback closes the loop.Source: A. MIMO Formulation of Grid-Forming… · p. 3
• The proposed generalized architecture closes the loop with a 3 × 5 control transfer matrix Φ and error signals that may be linear or nonlinear.The nominal frequency reference is set to the grid frequency ωg.Source: A. MIMO Formulation of Grid-Forming… · p. 3

1) Droop Control:

• Droop control emulates synchronous-generator regulation droop characteristics,.Source: 1) Droop Control · p. 3
• The general droop strategy assumes an ideal DC source and neglects DC control.Source: 1) Droop Control · pp. 3–4
• In practice, a PI controller is usually used to control the DC-capacitor voltage.Source: 1) Droop Control · p. 4
• The generalized droop implementation is shown as a control block in Fig. 5 and expressed through a control transfer matrix.Source: 1) Droop Control · p. 4, Figure 5 · p. 4
• Its transfer matrix uses one SISO PI controller and two SISO proportional controllers, with no coupling between rows or columns.Source: 1) Droop Control · p. 4
• Because the matrix is uncoupled, it typically omits real coupling among vdc, p, ωu, q, and V.Source: 1) Droop Control · p. 4

2) PSC Control:

• PSC directly emulates the synchronous-generator relationship between rotor angle and active power.Source: 2) PSC Control · p. 4
• PSC is equivalent to droop with respect to ωu, and its DC- and AC-voltage controls can match droop, giving it the same transfer-matrix structure.Source: 2) PSC Control · p. 4

3) VSG Control:

• VSG control emulates synchronous-generator behavior more fully than droop or PSC control,,.The paper presents a basic VSG architecture in the generalized configuration.Source: 3) VSG Control · p. 4
• In VSG control, inertia constant H emulates inertia and damping coefficient kp emulates damping; DC control usually uses a PI controller.Its transfer matrix also uses PI controllers, low-pass filters, and I controllers to derive control inputs, without coupling among DC-voltage, AC-voltage, and frequency-control rows.Source: 3) VSG Control · p. 4

4) Matching Control:

• Matching control generates ωu from vdc because DC-capacitor dynamics are similar to the synchronous-generator rotor.The paper derives its mathematical model and represents it as an equivalent block in the generalized configuration.Source: 4) Matching Control · p. 4
• The matching-control transfer matrix uses two P controllers and one PI controller, with the first two rows coupled through the DC signal..Source: 4) Matching Control · p. 4
• The analysis shows that basic grid-forming controllers can be transformed into the generalized configuration as particular control transfer matrices.Analogous reasoning extends this representation to various improved formulations.Source: 4) Matching Control · p. 4
• Table I summarizes the control transfer matrices used by different grid-forming controllers.Source: 4) Matching Control · p. 4, Table I · p. 5

C. Discussion on Proposed Generalized Configuration

• The configuration represents basic and improved grid-forming controls as transfer matrices, making their comparisons and relationships explicit.For example, PSC-2, PSC-3, and VSG-8 have identical frequency controls; droop-5 and VSG-2 are identical for output-side disturbances but not input-side disturbances, despite equivalence claims in,.Source: C. Discussion on Proposed Generalized… · p. 5
• Jointly optimizing DC, active-power, and reactive-power loops can exploit coupling terms to improve performance.This contrasts with the decoupling used by most existing controls to simplify design and analysis.Source: C. Discussion on Proposed Generalized… · p. 5
• New controls can use nonlinear elements or add coupling terms to the transfer matrix.The paper suggests sine-function controllers for global stability as in [7], and DC-signal coupling motivated by matching control [34].Source: C. Discussion on Proposed Generalized… · p. 6
• The framework can incorporate virtual impedance and inner voltage and current loops, and can replace those SISO loops with multivariable design.It can also use error signals beyond e = Y_ref − y, including the PSC and dVOC examples in and.Source: C. Discussion on Proposed Generalized… · p. 6
• Fixed control transfer matrices can be generalized to time-varying or grid-impedance-angle-dependent forms to encompass adaptive control.Source: C. Discussion on Proposed Generalized… · p. 6

A. Proposed Control Transfer Matrix

• The proposed matrix adds coupling terms to improve on existing sparse transfer matrices based on SISO loops.Source: A. Proposed Control Transfer Matrix · p. 6
• The φ11 PI controller regulates vdc with zero steady-state error, and kidc may be set to zero for matching controllers.Source: A. Proposed Control Transfer Matrix · p. 6
• The φ22 controller provides frequency-control inertia while maintaining the prescribed steady-state droop characteristics.Source: A. Proposed Control Transfer Matrix · p. 6
• Because a PLL is not expected, the third matrix column is zero; other choices are possible when a PLL is available.Source: A. Proposed Control Transfer Matrix · p. 6
• An integral controller maintains the steady-state q-V droop characteristics.Source: A. Proposed Control Transfer Matrix · p. 6
• Coupling terms from q and V into frequency control are chosen not to change the steady-state p-f droop characteristics.Source: A. Proposed Control Transfer Matrix · p. 6
• The remaining elements are P controllers, so the added coupling terms do not increase Φ's order relative to basic droop control with LPFs and VSG.Source: A. Proposed Control Transfer Matrix · p. 6

1) Formulation of H∞synthesis:

• Intermediate vectors ˆu and ˆy separate the control parameters to recast parameter design as a standard H-infinity optimization problem.Fig. 8 depicts the control transfer matrix used for this separation.Source: 1) Formulation of H∞synthesis · p. 6, Figure 8 · p. 7
• All parameters to be designed are contained in the static gain K.Source: 1) Formulation of H∞synthesis · p. 6
• The standard linear fractional transformation represents the system as G apart from K, with w as disturbance inputs and z as performance outputs.Fig. 9 shows the converter in this representation; the passage does not specify the particular choices of w and z.Source: 1) Formulation of H∞synthesis · p. 6, Figure 9 · p. 7

• The H∞ optimization uses weighting functions W_ij(s) to limit disturbance responses from w_j to z_i through transfer functions T_ij(s).Source: 2) Choices of Weighting Functions · p. 7
• W_11(s) is chosen to keep the low-frequency gain of T_11(s) small, supporting a small active-power tracking error.Source: 2) Choices of Weighting Functions · p. 7
• W_21(s) and W_22(s) promote rapid high-frequency decay in T_21(s) and T_22(s) to limit active-power disturbance, while W_31(s) and W_32(s) limit high-frequency gains so ω_u does not vary quickly.Source: 2) Choices of Weighting Functions · p. 7
• The weighting-function numbers are summarized in Table II and chosen based on converter parameters; Fig. 10 shows their log-magnitude curves.The gain vector K is derived by solving the optimization, for example with Matlab’s hinfstruct.Source: Table II · p. 7, 2) Choices of Weighting Functions · p. 7, Figure 10 · p. 7
• W_41(s) limits the low-frequency gain of T_41(s) so control of ω_u does not affect steady-state q-V droop regulation.Source: 2) Choices of Weighting Functions · p. 7

A. Simulation results

• The time-domain comparisons use Matlab/Simulink simulations with the converter topology of Fig. 1 and parameters listed in Table III.The passage identifies the parameter table but does not provide its values.Source: A. Simulation results · p. 7
• The proposed control’s H∞ optimization uses Matlab’s hinfstruct routine, initialized with K = diag(90,400,0,0,0,20,0,0,0,0,0,1,1).The supplied passage does not state the resulting optimized gains.Source: A. Simulation results · p. 7
• For comparison, VSG-2’s sparse control transfer matrix is designed using the same optimization routine as the proposed controller.The comparison is intended to show the benefits of the added coupling terms.Source: A. Simulation results · p. 8
• Droop-5 parameters are designed through their equivalent relationship with VSG-2, as described in and.Source: A. Simulation results · p. 8
• MIMO-GFM damps oscillations after Pref steps from 0.5 p.u. to 1 p.u., while droop-5 and VSG-2 show larger oscillations and less smooth dynamics.The different droop-5 and VSG-2 responses arise because Pref is treated as an input-side disturbance.Source: A. Simulation results · p. 8
• When grid frequency decreases from 50 Hz to 49.9 Hz, MIMO-GFM synchronizes quickly and smoothly with almost no overshoot, unlike the larger oscillations under droop-5 and VSG-2.In this case, droop-5 and VSG-2 responses are equivalent because grid frequency is treated as an output-side disturbance.Source: A. Simulation results · p. 8

B. Experimental results

• The experiments used a Danfoss drive, LCL filter, Chroma grid simulator, and dSPACE control system; fixed DC-source control made the DC-voltage row unavailable as a design variable.The experimental setup's parameters matched Table III; the frequency- and voltage-control coupling terms remained demonstrable.Source: B. Experimental results · p. 8, Figure 13 · p. 9

V. CONCLUSION

• The generalized configuration treats DC-voltage, frequency, and voltage control as one MIMO transfer matrix to be designed, rather than assuming decoupled loops.Source: §V · p. 9
• The configuration unifies popular grid-forming controls and their improved formulations in a generalized control transfer matrix.Source: §V · p. 9

B. Experimental results

• The proposed MIMO-GFM responds to Pref and grid-frequency changes quickly and smoothly with almost no oscillations, while traditional VSG control could not be stabilized experimentally.VSG oscillations could be damped in simulations, but parameter errors, delays, and other physical-system disturbances prevented stabilization in the experiments.Source: B. Experimental results · p. 8, Figure 14 · p. 9, Figure 15 · p. 9

V. CONCLUSION

• The generalized configuration supports control comparison and design, and the proposed new MIMO-GFM controller does not increase controller order.Source: §V · p. 9
• The paper transforms optimal control design with multiple parameters into a standard H∞ optimization problem.The paper reports that simulation and experimental results verify the proposed method's superior performance.Source: §V · p. 9