MULTI-OPERATIONAL MODE INVERTERS THAT FACILITATE A UNIFIED CONTROL DESIGN ACROSS DIFFERENT POWER GRID TYPES

Abstract

A computing system for functional integration with a distributed power grid having one or more inverters. The computing system includes processing device(s) to control operation of an inverter by executing a feedback controller with a grid-tied line, of the distributed power grid, treated as a plant and including a unified algebraic control system operating based on a pair of closed-loop variables carried in closed-loop signals of the unified algebraic control system. The processing device(s) control, using the feedback controller, transitions of the inverter between a plurality of operating modes by adjusting a magnitude of the closed-loop signals, which correspond to the pair of closed-loop variables, towards a setpoint of a plurality of setpoints. Each setpoint corresponds to a different operating mode of the plurality of operating modes.

Claims

1. A computing system for functional integration with a distributed power grid comprising one or more inverters, wherein the computing system comprises: one or more processing devices; and memory communicatively coupled with and readable by the one or more processing devices and having stored therein processor-readable instructions which, when executed by the one or more processing devices, cause the one or more processing devices to perform operations comprising: controlling operation of an inverter, of the one or more inverters, by executing a feedback controller with a grid-tied line, of the distributed power grid, treated as a plant and comprising a unified algebraic control system operating based on a pair of closed-loop variables carried in closed-loop signals of the unified algebraic control system; and controlling, using the feedback controller, transitions of the inverter between a plurality of operating modes by adjusting a magnitude of the closed-loop signals, which correspond to the pair of closed-loop variables, towards a setpoint of a plurality of setpoints, wherein each setpoint corresponds to a different operating mode of the plurality of operating modes.

2. The computing system of claim 1, wherein the plurality of operating modes comprises at least two or more of grid-forming (GFM), grid-following (GFL), static synchronous compensator (STATCOM), energy storage system (ESS), or voltage source inverter (VSI).

3. The computing system of claim 1, wherein the controlling the transitions further comprises adjusting the pair of closed-loop variables along trajectories in a control parameter space, starting and ending at setpoints, of the plurality of setpoints, corresponding respectively to an initial mode and a final mode of the plurality of operating modes.

4. The computing system of claim 1, wherein the pair of closed-loop variables comprise an inverter variable and a grid state variable, which are implemented as a control input and a disturbance using a direct quadrature formulation.

5. The computing system of claim 1, wherein controlling the transitions further comprises: modeling, using direct quadrature formulation, inputs and outputs of the unified algebraic control system using a pair of a single input, single output (SISO) models comprising sensitivity and a multiplicative perturbation; and deriving a coupling between the pair of SISO models.

6. The computing system of claim 5, wherein the operations further comprise: factoring a set of primary closed-loop models and a corresponding closed-loop sensitivity model into a set of diagonal parts; determining a bounded coupling perturbation between the set of diagonal parts; and shaping sensitivity of the set of diagonal parts to minimize the bounded coupling perturbation.

7. The computing system of claim 1, wherein the operations further comprise: quantifying sensitivities of a pair of two q-axis transfer functions of the unified algebraic control system, wherein the pair of two q-axis transfer functions comprises: i) a closed-loop transfer function between a current setpoint and a grid current; and ii) a transient frequency response of the inverter; generating a measure of robustness to model perturbations as a product of the sensitivities; and bounding a frequency range of operation of the inverter according to bounds of the measure of robustness to preserve stability and performance characteristics under plant perturbations.

8. The computing system of claim 1, wherein the operations further comprise: implementing, as part of the unified algebraic control system, a cascaded closed-loop structure with an inner current loop and an outer voltage loop; using feedback linearization to decouple inductor dynamics of an inductor and a capacitor being modeled within the inner current loop; and employing a phase interpolator controller to shape the inner current loop into a unity gain low-pass filter.

9. The computing system of claim 1, wherein the feedback controller comprises a multiple input, multiple output (MIMO) feedback controller decomposed into: a MIMO plant shaping controller component configured to transform line dynamics into a modified plant with reduced coupling and cross-channel interaction between a d-axis for voltage and a q-axis for frequency; and a diagonal controller component configured to maintain stability and achieve target objectives associated with a decoupled d-axis and a decoupled q-axis under an impact of the modified plant.

10. The computing system of claim 9, wherein the plant shaping controller component is configured to achieve at least one of: diagonal modified plant dynamics at steady state; triangular modified plant dynamics at steady state; or an optimal condition number for the modified plant across frequency ranges for a highest stability margin.

11. The computing system of claim 1, wherein the operations further comprise implementing mode-independent synchronization by maintaining synchronization conditions across the plurality of operating modes by: ensuring capacitor voltage alignment with a direct-quadrature (dq) reference frame; and achieving a frequency lock between the inverter and a grid frequency.

12. The computing system of claim 11, wherein the synchronization is achieved by configuring: a first q-axis controller with at least two zeros at the origin relative to a second q-axis controller; and the second q-axis controller with at least one pole at the origin.

13. The computing system of claim 1, wherein the operations further comprise characterizing the plurality of operating modes based on a number of zeros at the origin in sensitivity transfer functions, wherein characterizing comprises at least two of: a grid-following mode corresponds to one zero in a d-axis, associated with grid voltage, and two zeros in a q-axis associated with grid frequency; a grid-forming mode corresponds to no zeros in the d-axis and one zero in the q-axis; a static synchronous compensator (STATCOM) mode corresponds to no zeros in the d-axis and two zeros in the q-axis; or an energy storage system (ESS) mode corresponds to one zero in the d-axis and one zero in the q-axis.

14. The computing system of claim 1, wherein the operations further comprise implementing virtual inertia control by shaping a frequency response transfer function as a low-pass filter with adjustable bandwidth and damping characteristics to control: a rate of change of frequency between the inverter and the grid-tied line; and frequency nadir following disturbances bypassing a reliance on mimicking synchronous generator dynamics.

15. The computing system of claim 14, wherein the operations further comprise deriving parameters for the virtual inertia control from a surrogate second-order system model relating: a phase margin to system damping; and a gain crossover frequency to virtual inertia constant.

16. The computing system of claim 1, wherein the operations further comprise implementing droop characteristics for power sharing among parallel inverters, of the one or more inverters, wherein: d-axis droop coefficients determine voltage regulation characteristics, wherein the d-axis droop coefficients are bounded by line impedance characteristics for d-axis voltage droop; and q-axis droop coefficients, associated with grid frequency, determine frequency regulation characteristics, wherein the q-axis droop coefficients are bounded by a targeted frequency response bandwidth for q-axis frequency droop.

17. The computing system of claim 1, wherein the operations further comprise implementing current limiting operation by dynamically adjusting operating mode parameters using a barrier function based on a ratio of output current to a maximum allowable current, wherein the barrier function causes transition toward a grid-following mode as grid current approaches the maximum allowable current.

18. The computing system of claim 17, wherein the barrier function comprises a logarithmic barrier that scales operating mode parameters inversely with proximity to the maximum allowable current.

19. The computing system of claim 1, wherein the operations further comprise compensating for line-to-ground fault conditions by incorporating proportional-resonant compensators tuned to second harmonic frequency to maintain current control during fault conditions.

20. A method for controlling distributed inverters in a multi-source power grid, the method comprising: executing, by a computing system operatively coupled to the distributed inverters, a unified algebraic control system in which different operating modes of the distributed inverters are defined by magnitudes of closed-loop signals, wherein the different operating modes cause differing control of the distributed inverters, and the closed-loop signals are associated with a plurality of control parameters; parameterizing, within the unified algebraic control system, an operational space of the multi-source power grid as a two-dimensional continuum of the different operating modes using the plurality of control parameters; enabling smooth transitions between the operating modes by adjusting the plurality of control parameters along trajectories in the operational space between setpoints of the plurality of control parameters; and maintaining, by the computing system, stability during inverter transitions through pointwise stability at each operating mode, of the operating modes, and a controlled rate of parameter change associated with the plurality of control parameters.

21. The method of claim 20, further comprising: factoring multiple input, multiple output (MIMO) sensitivity into: diagonal two-single input, single output (2-SISO) sensitivity components; and multiplicative coupled perturbation terms; wherein the factorization enables conversion of MIMO analysis into simpler SISO problems than the MIMO analysis.

22. The method of claim 20, further comprising implementing a plant shaping controller that satisfies: decoupling conditions at steady state through triangular or diagonal structure; stability margin optimization at high frequencies; and condition number optimization for robustness to input uncertainty.

23. The method of claim 20, further comprising designing, within the unified algebraic control system, a d-axis controller and a q-axis controller with specified characteristics comprising at least one of: the d-axis controller determines voltage regulation and reactive power support; the q-axis controller comprises parallel branches for frequency response and synchronization; or the plurality of control parameters are selected to achieve a target bandwidth, a target phase margin, and a target steady-state droop.

24. The method of claim 20, further comprising implementing a seamless transition between grid-connected and islanded operation without implementing control reconfiguration, islanding detection schemes, or communication-based coordination.

25. A non-transitory computer-readable storage medium storing instructions that, when executed by one or more processors, cause the one or more processors to perform operations for inverter control, the operations comprising: executing a feedback control algorithm that treats power grid connection of one or more inverters, of a multi-source power grid, as a plant comprising resistor-inductor impedance; maintaining, associated with the feedback control algorithm, a two-dimensional operating space parameterized by voltage and frequency control variables associated with the multi-source power grid; causing, using the feedback control algorithm, transitions between a plurality of operating modes of the one or more inverters by adjusting the voltage and frequency control variables according to grid conditions and power requirements of the multi-source power grid; and ensuring stability of operation of the one or more inverters within the multi-source power grid through satisfaction of sector-boundedness conditions for nonlinear dynamics modeled within the feedback control algorithm.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0003] A more particular description of the disclosure briefly described above will be rendered by reference to the appended drawings. Understanding that these drawings only provide information concerning typical embodiments and are not therefore to be considered limiting of its scope, the disclosure will be described and explained with additional specificity and detail through the use of the accompanying drawings.

[0004] FIG. 1 is an operational schematic diagram illustrating a collection of inverter units to assimilate distributed energy resource and storage units into a power grid according to some embodiments.

[0005] FIG. 2A is a circuit diagram illustrating an inverter as a controlled voltage source, v.sub.c, connected to the power grid, v.sub.g, via an resistor-inductor (RL) impedance according to various embodiments.

[0006] FIG. 2B is a graph of dq synchronous rotating frame associated with FIG. 2A according to some embodiments.

[0007] FIG. 3 is a circuit diagram illustrating a proposed closed-loop with line dynamics G.sub.L in Equation (4) as plant and K={tilde over (K)}K.sub.L as a multiple input, multiple output (MIMO) feedback controller according to some embodiments.

[0008] FIG. 4A, FIG. 4B, FIG. 4C, FIG. 4D are Bode plots illustrating inverter operating mode based on the shape and number of zeros in S at low-frequency in, respectively: (a) grid-forming (GFM); (b) voltage support in static synchronous compensator (STATCOM); (c) frequency support in an energy storage system (ESS); and (d) grid-following (GFL).

[0009] FIG. 5 is a continuum representation of a spectrum between GFL and VSI parameterized by {tilde over (S)} according to some embodiments.

[0010] FIG. 6 is a Bode plot illustrating operations of desired q-axis closed-loops T.sub.v, T.sub., {tilde over (T)}.sup.q, and corresponding open-loops

[00001]$K_1^q{\stackrel{}{G}}_M,K_2^q{\stackrel{}{G}}_M$

ana K.sup.q{tilde over (G)}.sub.M according to some embodiments.

[0011] FIG. 7 is a Bode plot illustrating smaller values of , which impose stricter upper-bounds on sensitivity around the fundamental frequency .sub. according to at least one embodiment.

[0012] FIG. 8A and FIG. 8B are circuit diagrams illustrating, respectively, (a) nested structure with voltage and current inputs; and (b) inverter closed-loop as part of feedback compensator.

[0013] FIG. 9 is multi-dimensional graph illustrating operating space of a multi-operational mode power grid parameterized by (.sub.v, .sub.) according to some embodiments.

[0014] FIG. 10A, FIG. 10B, FIG. 10C, and FIG. 10D are graphs of current over time during nominal grid operation, respectively: (a) active current with and without set-point adjustment; (b) reactive current in a complex line scenario; (c) active current with adjusted current set-point; and (d) reactive current in inductive line scenario according to various embodiments.

[0015] FIG. 11A, FIG. 11B, FIG. 11C, FIG. 11D are graphs of current over time during disturbed grid operation, respectively: (a) active current with and without set-point adjustment under grid frequency disturbance; (b) reactive current deviation under grid volage disturbance; (c) improving active current with adjusted current set-point; and (d) improving reactive current in inductive line scenario by decreasing the sensitivity transfer function according to various embodiments.

[0016] FIG. 12 is a graph of a frequency response of the inverter to step change of magnitude 0.1 hertz (Hz) in grid frequency according to at least one embodiment.

[0017] FIG. 13 is a set of graphs illustrating how a proposed GFM control system effectively rejects a 120 Hz ripple caused by line-to-ground fault on inverter current and frequency according at least one embodiment.

[0018] FIG. 14A, FIG. 14B, FIG. 14C, FIG. 14D are graphs illustrating, respectively, d-axis current, q-axis current, inverter voltage, and inverter frequency of a seamless transition of the set of inverters during on-grid operation at particular moments outline in Table III according to some embodiments.

[0019] FIG. 15A, FIG. 15B, FIG. 15C are graphs illustrating, respectively, (a) a Bode plot for d-axis and q-axis open-loops under GFM and GFL operation; (b) GFM response to step-change in active current set-point under line impedance variation; and (c) GFL response to step-change in active current set-point under line impedance variation according to some embodiments.

[0020] FIG. 16A, FIG. 16B, FIG. 16C, FIG. 16D are graphs illustrating, respectively, d-axis current, q-axis current, inverter voltage, and inverter frequency of a seamless transition of the set of inverters during on-grid operation at particular moments outline in Table IV according to some embodiments.

[0021] FIG. 17A is a control diagram of a three-phase phase-locked loop (PLL) system for GFL inverters according to some embodiments.

[0022] FIG. 17B is a control diagram of modified frequency control for GFN inverters to mimic rotor dynamic and governor action according to some embodiments.

[0023] FIG. 18 is a graph of Bode plots for five key transfer functions in the q-axis and their corresponding open-loop transfer functions according to some embodiments.

[0024] FIG. 19A is a circuit diagram of a three-phase DC/AC inverter with sensor measurements for feedback control according to some embodiments.

[0025] FIG. 19B is a control diagram representation of the embedded implementation of the feedback controller according to some embodiments.

[0026] FIG. 20A is a graph of an inverter's output current magnitude as a function of grid voltage deviation (v) and frequency deviation () according to an embodiment.

[0027] FIG. 20B is a 2D cross-section of the surface in FIG. 20A along the i.sup.q=0 plane according to an embodiment.

[0028] FIG. 20C is a 2D cross-section of the surface in FIG. 20A along the i.sup.d=0 plane according to an embodiment.

[0029] FIG. 21 illustrates a block diagram illustrating an exemplary computer device (or computing device), in accordance with implementations of the present disclosure.

[0030] While embodiments of the present disclosure are susceptible to various modifications and alternative forms, exemplary embodiments thereof are shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the description of exemplary embodiments is not intended to limit the disclosure to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the in accordance with one or more embodiments of the present disclosure.

DETAILED DESCRIPTION

[0031] The technology now will be described more fully hereinafter with reference to the accompanying drawings, in which some, but not all embodiments are shown. Indeed, the disclosure may be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will satisfy applicable legal requirements.

[0032] Likewise, many modifications and other embodiments of the technology described herein will come to mind to one of skill in the art to which the disclosure pertains having the benefit of the teachings presented in the enclosed descriptions and the associated drawings. Therefore, it is to be understood that the disclosure is not to be limited to the specific embodiments disclosed and that modifications and other embodiments are intended to be included within the scope of this disclosure. Although specific terms are employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation.

[0033] Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of skill in the art to which embodiments described herein pertain. Although any methods and materials similar to or equivalent to those described herein can be used in the practice or testing of embodiments of the present disclosure, the preferred methods and materials are described herein.

[0034] Existing control frameworks for direct-current (DC)/alternating-current (AC) power inverters are significantly customized and fine-tuned to a specific and fixed operating mode based on their intended use, limiting their versatility. For example, in the presence of a stiff grid, inverters operate in grid-following (GFL) mode, where they synchronize and supply power to the grid. On the other hand, in the absence of a viable or stiff grid, inverters usually operate in grid-forming (GFM) mode, where they not only control the power flow, but also provide voltage and frequency support at the point of common coupling (PCC). The other popular operating modes are the static synchronous compensator (STATCOM) and energy storage system (ESS), where the inverters serve as ancillary devices (grid support).

[0035] In STATCOM mode, the inverter improves voltage regulation and power transfer capacity by balancing reactive power. In contrast, an ESS acts as a rapidly dispatchable power source, primarily enhancing frequency control by exchanging active power with the connected AC system. The existing literature focuses predominantly on designing distinct control systems to address the objectives of each mode separately. However, with the increased integration of uncertain DERs and highly dynamic power consumption and generation profiles, it is becoming critical for the same inverter to operate in different operational modes at different times. In this context, the inverter control framework should facilitate a seamless transition between operating modes. For example, when most inverters operate in GFL mode with Maximum Power Point Tracking (MPPT), rapid spatio-temporal changes in irradiance (e.g., irradiance anomalies due to moving clouds) lead to rolling and non-localized power imbalance in the network. To address this issue, transitioning some of the inverters near the impacted MPPT areas to ESS or GFM can help locally adjust the power and prevent propagation of undesired voltage and frequency transients in the network.

[0036] FIG. 1 is an operational schematic diagram illustrating a collection of inverter units to assimilate distributed energy resource and storage units 110 into a power grid 120 according to some embodiments. In embodiments, the energy resource and storage units 110 include batteries 110A, electric vehicles 110B, renewables 110C, and a generator set 110D that feed the power grid 120. The generator set 110D (also referred to as gen set) can provide a non-renewable source of power. Another common scenario involves transitioning between grid-tied (GFL) and islanded (GFM) modes, particularly during grid fault. This transition can be facilitated by reconfiguring and switching an inverter controller 130 based on islanding detection schemes, e.g., via control logic 132. In some embodiments, the inverter controller 130 is implemented by a computing device 2100 or computing system such as discussed with reference to FIG. 21. However, most islanding detection methods rely on real-time, centralized communication, making them vulnerable to communication delays and blackouts, which limits the robustness of the microgrid. Furthermore, sudden switching and reconfiguration of controllers can result in harmful transient response and instability in low-inertia inverter-based microgrids during the transition. Consequently, there has been increasing research on achieving seamless transition in weak-grid conditions without control reconfiguration and islanding detection.

[0037] Low-inertia refers to portions of a distributed and mix-source power grid lacking large synchronous generators (SG) that retain stored kinetic energy. This stored kinetic energy within such generators can act as a buffer to provide inertia when demand suddenly fluctuates. Microgrids made up of renewal power sources such as solar photovoltaic, wind turbines, batteries, and the like, are usually connected to the grid through power electronics, e.g., power inverters referred to here more generally as inverters. These renewal power sources, however, lack the natural inertia of large SG.

[0038] Synchronverters, which emulate the behavior of SG, offer a promising approach to seamless transition. These systems replace conventional voltage and current loops with angle, frequency, and torque controllers, providing emulated inertia and enhanced frequency dynamics. However, relying solely on SG dynamics may not fully exploit the fast dynamics of the inverters or allow for more advanced control strategies, potentially resulting in limited stability margins and suboptimal performance. Consequently, numerous studies have focused on improving stability margins, damping characteristics, and reducing sensitivity to weak grid coupling by leveraging inverter dynamics.

[0039] Another approach involves the perpetual operation of the inverters in droop-based grid-forming mode regardless of grid availability. These methods propose dynamically improved droop laws to maintain system stability in both grid-connected and islanded modes. Control strategies based on multivariate loop-shaping techniques are explored to decouple active and reactive power in weak grids. Although systematic, these approaches lack multiple-input multiple-output (MIMO) stability analysis, and performance analysis that generalizes beyond the proposed loop-shaping controller. At least one work provides a mode-dependent droop-based GFM that can operate in a grid-tied mode and improve reactive power regulation upon receiving a mode switch signal.

[0040] Aspects of the present disclosure address the above and other deficiencies of present multi-source power grid approaches through employing an integrated control system designed to achieve seamless transitions among a spectrum of inverter operational modes according to various embodiments. The operation spectrum can include, for example, grid-forming (GFM), grid-following (GFL), static synchronous compensator (STATCOM), energy storage system (ESS), and voltage source inverter (VSI). The proposed control architecture offers guarantees of stability, robustness, and performance regardless of the specific mode.

[0041] In some embodiments, the disclosed systems and methods establish a unified algebraic structure for the feedback control system, where different modes are defined by the magnitude of closed-loop signals. As demonstrated, this approach results in a two-dimensional continuum of operational modes and enables transition trajectories between operational modes by dynamically adjusting closed-loop variables towards corresponding setpoints. Stability, robustness, and fundamental limitation analyses are provided for the closed-loop system across any mode, as well as during transitions between modes. This design facilitates stable and enhanced on-grid integration, even during GFM operation and weak grid conditions. Ultimately, attributes of the proposed system are demonstrated through simulations and experiments designed to highlight its efficacy in managing on-grid operation and islanding scenarios, particularly in situations of weak grid operation. This disclosure provides extensive analysis by modeling the GFM in both on-grid and off-grid operations and investigating the root locus for a range of parameters.

[0042] In various embodiments, the disclosed control system solves the multi-mode operation challenge through three aspects that work together like layers of a control hierarchy. First, imagine a power inverter as a translator between DC and AC power that needs to speak different languages (e.g., control formulation) depending on needs of the power grid. In strong grid conditions, the power inverter acts like a current source (GFL mode), precisely following power commands. In weak or absent grid conditions, the power inverter acts like a voltage source (GFM mode), maintaining stable voltage and frequency. In some embodiments, therefore, the power inverter creates one unified translator that allows smooth translation between these languages. Second, the control system may simplify control by separating complex coupled dynamics (where voltage affects frequency and vice versa) into independent channels. Third, the system includes built-in safeguards against real-world uncertainties, e.g., variations in line impedance, measurement noise, and grid disturbances. These safeguards work like shock absorbers in a car, maintaining stable operation even when the road (grid conditions) becomes rough. The following disclosure details how these concepts are implemented mathematically and practically.

[0043] In some embodiments, a computing system is disclosed for functional integration with a distributed power grid having one or more inverters, e.g., inverters that control power flow between multiple power sources (such as DERs) in a multi-source ground. The computing system can include one or more processing devices and memory communicatively coupled with and readable by the one or more processing devices and having stored therein processor-readable instructions which, when executed by the one or more processing devices, cause the one or more processing devices to perform operations disclosed herein below. The operations, for example, can include controlling operation of an inverter, of the one or more inverters, by executing a feedback controller with a grid-tied line, of the distributed power grid, treated as a plant. The feedback controller can further include a unified algebraic control system operating based on a pair of closed-loop variables carried in closed-loop signals of the unified algebraic control system. The operations further include controlling, using the feedback controller, transitions of the inverter between a plurality of operating modes by adjusting a magnitude of the closed-loop signals, which correspond to the pair of closed-loop variables, towards a setpoint of a plurality of setpoints. In embodiments, each setpoint corresponds to a different operating mode of the plurality of operating modes.

[0044] The operations can further include quantifying sensitivities of a pair of two q-axis transfer functions of the unified algebraic control system. In embodiments, the pair of two q-axis transfer functions include: i) a closed-loop transfer function between a current setpoint and a grid current; and ii) a transient frequency response of the inverter. The operation can include generating a measure of robustness to model perturbations as a product of the sensitivities and bounding a frequency range of operation of the inverter according to bounds of the measure of robustness to preserve stability and performance characteristics under plant perturbations.

[0045] In related embodiments, a method for controlling distributed inverters in a multi-source power grid includes executing, by a computing system operatively coupled to the distributed inverters, a unified algebraic control system in which different operating modes of the distributed inverters are defined by magnitudes of closed-loop signals, wherein the different operating modes cause differing control of the distributed inverters, and the closed-loop signals are associated with a plurality of control parameters. The method can include parameterizing, within the unified algebraic control system, an operational space of the multi-source power grid as a two-dimensional continuum of the different operating modes using the plurality of control parameters. The method can include enabling smooth transitions between the operating modes by adjusting the plurality of control parameters along trajectories in the operational space between setpoints of the plurality of control parameters. The method can include maintaining, by the computing system, stability during inverter transitions through pointwise stability at each operating mode, of the operating modes, and a controlled rate of parameter change associated with the plurality of control parameters.

[0046] In some embodiments, the method further includes designing, within the unified algebraic control system, a d-axis controller and a q-axis controller with specified characteristics, These characteristics can include: the d-axis controller determines voltage regulation and reactive power support; the q-axis controller comprises parallel branches for frequency response and synchronization; and/or the plurality of control parameters are selected to achieve a target bandwidth, a target phase margin, and a target steady-state droop. The method can further include implementing a seamless transition between grid-connected and islanded operation without implementing control reconfiguration, islanding detection schemes, or communication-based coordination.

[0047] Therefore, advantages of the systems and methods implemented in accordance with some embodiments of the present disclosure include, but are not limited to, a unified control system that facilitates seamless transitions between different operating modes. Included in this approach is establishing a common algebraic system or framework for the feedback control system across various modes. In this structure, the operational space of the inverter can be viewed as a continuum of modes, including GFM, GFL, STATCOM, and ESS. Each mode can be parameterized by two control variables. Smooth transitions between modes can be enabled by adjusting these parameters along trajectories in the parameter space, starting and ending at setpoints corresponding to the initial and final modes. The proposed system transforms inverter control objectives into closed-loop properties of the feedback control structure. This approach inherently incorporates the benefits of existing state-of-the-art design concepts such as droop, virtual impedance, PLL, and inertia, without explicitly designing for them. In some embodiments, the present feedback control structure results in a universal characterization of stability, performance, and fundamental trade-offs for the MIMO closed-loop, regardless of the operational mode. Other advantages will be apparent to those skilled in the art of power semiconductor transistor design and fabrication, which will be discussed hereinafter.

[0048] FIG. 2A is a circuit diagram illustrating an inverter as a controlled voltage source, v.sub.c, connected to the power grid, v.sub.g, via an resistor-inductor (RL) impedance according to various embodiments. FIG. 2B is a graph of dq synchronous rotating frame associated with FIG. 2A according to some embodiments. In the disclosed system, the inverter interface can be associated with a power network, which is common to all operational modes, as two voltage sources connected with an RL impedance as shown in FIG. 2A. Here, v.sub.c is the inverter's capacitor voltage, while v.sub.g represents the PCC voltage governed by the rest of the microgrid, i.e., the power grid and/or other inverter units. The RL impedance signifies the transmission line impedance R.sub.g+jL.sub.g and the grid side impedance of the LCL filter R.sub.f+jL.sub.f (R.sub.f, L.sub.f=0 in case of LC filter). Regardless of the mode of operation, it may be desired that the inverter in FIG. 2A maintains v.sub.c close to the nominal value of v.sub.0 while regulating the active and reactive power-flow {P, Q} through the impedance close to a {P.sub.0, Q.sub.0} set-point. The steady-state power-flow at fundamental frequency can be expressed as

[00002]$\begin{array}{cc}P=\left(\frac{.Math.v_c.Math..Math.v_g.Math.}{Z}\cos -\frac{.Math.v_g^2.Math.}{Z}\right)\cos {}_z-\frac{.Math.v_C.Math..Math.v_G.Math.}{Z}\sin {}_z\sin ,& \left(1\right)\end{array}$$Q=\left(\frac{.Math.v_c.Math..Math.v_g.Math.}{Z}\cos -\frac{.Math.v_g^2.Math.}{Z}\right)\sin {}_z+\frac{.Math.v_C.Math..Math.v_G.Math.}{Z}\cos {}_z\sin ,$

where is the phase difference between v.sub.c and v.sub.g and Z.sub.z:=R+jL in FIG. 2A. Different operating modes balance trade-offs between deviations in inverter output power and voltage from their nominal setpoints. For example, GFM and GFL inverters differ in the allowable voltage magnitude and frequency deviations to accommodate accurate tracking of the power reference, as set forth in Equation (1).

[0049] In some embodiments, the fundamental trade-offs between control objectives are characterized not only at steady-state, but throughout the frequency spectrum. Therefore, instead of relying on the nonlinear steady-state power flow equation in Equation (1) or the dynamic power phasor, exploited are the rich and linear dynamics of the grid current, i.sub.g, for the design and analysis of closed-loop control. In this context, the averaged dynamical model of the transmission line and inverter, as illustrated in FIG. 2A, in the direct quadrature (dq) frame can be expressed as

[00003]$\begin{array}{cc}L\frac{d\stackrel{.fwdarw.}{{\mathrm{.Math.}}_g}}{\mathrm{dt}}=\stackrel{.fwdarw.}{v_c}-\stackrel{.fwdarw.}{v_g}-R\stackrel{.fwdarw.}{{\mathrm{.Math.}}_g}-L\left({{}_0\left[-i_g^q,i_g^d\right]}^T+\left(t,\stackrel{.fwdarw.}{{\mathrm{.Math.}}_g}\right)\right)& \left(2\right)\end{array}$$\begin{array}{cc}{L}_i\frac{d\stackrel{.fwdarw.}{{\mathrm{.Math.}}_L}}{\mathrm{dt}}=\frac{v_{\mathrm{dc}}}{2}\stackrel{.fwdarw.}{m}-\stackrel{.fwdarw.}{v_c}-R_i\stackrel{.fwdarw.}{{\mathrm{.Math.}}_L}-L_i{\stackrel{}{}\left[-i_L^q,i_L^d\right]}^T,& \left(3\right)\end{array}$$C_i\frac{d\stackrel{.fwdarw.}{v_c}}{\mathrm{dt}}=\stackrel{.fwdarw.}{{\mathrm{.Math.}}_L}-\stackrel{.fwdarw.}{{\mathrm{.Math.}}_G}-C_i{\stackrel{}{}\left[-v_c^q,v_c^d\right]}^T.$

[0050] Here is the angular frequency of the dq frame, .sub.0 is the nominal frequency,

[00004]$\left(t,\underset{i_g}{.fwdarw.}\right)={\left(-{}_0\right)\left[-i_L^q,i_g^d\right]}^T$

is sector-bounded nonlinearity,

[00005]$\left(v_{\mathrm{dc}}/2\right)\underset{m}{.fwdarw.}$

is the cycle-averaged model of the switching node, and each vector contains the respective d and q signal components. Applying the Laplace transform to Equation (2) and Equation (3) results in

[00006]$\begin{array}{cc}\stackrel{.fwdarw.}{{\stackrel{}{\mathrm{.Math.}}}_g}=G_L\left({\stackrel{.fwdarw.}{\stackrel{}{v}}}_c-{\stackrel{.fwdarw.}{\stackrel{}{v}}}_g\right),\mathrm{where}{G}_L=\frac{\left[\begin{array}{cc}s+& {}_0\\ -{}_0& s+\end{array}\right]}{L\left(s^2+2s+{}^2+{}_0^2\right)}\mathrm{and}=\frac{R}{L},& \left(4\right)\end{array}$$\begin{array}{cc}{\stackrel{.fwdarw.}{\stackrel{}{v}}}_c=\frac{1}{C_{i}s}\left({\stackrel{.fwdarw.}{\stackrel{}{\mathrm{.Math.}}}}_L-{\stackrel{.fwdarw.}{\stackrel{}{\mathrm{.Math.}}}}_g-\left\{C_i{\left[-v_c^q,v_c^d\right]}^T\right\}\right),& \left(5\right)\end{array}$${\stackrel{.fwdarw.}{\stackrel{}{\mathrm{.Math.}}}}_L=\frac{1}{L_{i}s+R_i}\left(\frac{v_{\mathrm{dc}}}{2}\stackrel{.fwdarw.}{\stackrel{}{m}}-{\stackrel{.fwdarw.}{\stackrel{}{v}}}_c-\left\{L_i{\left[-i_L^q,i_L^d\right]}^T\right\}\right)$

where the hat notation signifies the Laplace domain signals, s is the Laplace variable, and {} is Laplace transform.

[0051] In some embodiments, the effect of nonlinearity (, ) is neglected in Equation (4). However, in Proposition 9.1 associated with Equation (75), sufficient conditions are established for the nominal closed-loop system under which the nonlinearity is sector-bounded and does not compromise closed-loop stability. Therefore, it is justifiable to neglect nonlinearity when the conditions outlined in Proposition 9.1 are met. Unlike the frequency-invariant input-output coupling in Equation (1), the frequency-dependent input-output coupling in Equation (4) allows design of controllers that achieve improved stability margins by exploiting the dynamic nature of the MIMO coupling.

[0052] FIG. 3 is a circuit diagram illustrating a proposed closed-loop with line dynamics G.sub.L in Equation (4) as plant and K={tilde over (K)}K.sub.L as a multiple input, multiple output (MIMO) feedback controller according to some embodiments. In embodiments, the K.sub.L is the full matrix part of K, while

[00007]$\tilde{K}=\mathrm{diag}\left(K^d,K_1^q+K_2^q\right)$

is the diagonal part of the controller. The input disturbances {d.sup.d, d.sup.q} capture the effect of the PCC on the closed-loop, while {u.sup.d, u.sup.q} integrates the capacitor voltage and the dq frame angle as closed-loop control effort.

[0053] In some embodiments, the closed-loop block diagram in FIG. 3 can be employed to analyze the stability and performance of the inverter and grid setup in FIG. 2A. In this context, G.sub.L represents the MIMO line dynamics in Equation (4) and the feedback controller K encapsulates the inverter dynamics in Equation (5). However, before the performance and stability are evaluated in terms of closed-loop in FIG. 3, one can resolve the control input {right arrow over (u)}=[u.sup.d, u.sup.q].sup. and disturbance {right arrow over (d)}=[d.sup.d, d.sup.q].sup. in terms of the inverter and grid state variables.

[0054] Proposition 3.1: For the closed-loop structure in FIG. 3, the control input {right arrow over (u)} and disturbance {right arrow over (d)} are

[00008]$\begin{array}{cc}\left[\begin{array}{c}{u}^d\\ u^q\end{array}\right]=\left[\begin{array}{c}{v}_c^d\\ v_c^q\end{array}\right]+\left[\begin{array}{c}0\\ u_{}\end{array}\right],\mathrm{and}\left[\begin{array}{c}{d}^d\\ d^q\end{array}\right]=\left[\begin{array}{c}{v}_g\\ .Math.v_g.Math.\mathrm{dt}\end{array}\right],& \left(6\right)\end{array}$

where

[00009]$v_c^d=v_c^d-v_0,v_g=.Math.v_g.Math.-v_0,v_c^q=v_c^q-0,u_{}\left(t\right)=.Math.v_g.Math.\mathrm{dt},=-{}_{},$

and .sub.g=.sub.g.sub.0. The control input u may represent what the inverter can actively control: the capacitor voltage deviation and the rotating reference frame angle. The disturbance d may represent what the inverter cannot control but responds to: grid voltage variations and grid frequency changes. These are the fundamental signals that determine whether the inverter maintains stable operation.

[0055] Proof 3.1: Base on FIG. 3 and Equation (4) result {right arrow over (v)}{right arrow over (d)}={right arrow over (v)}.sub.c{right arrow over (v)}.sub.g=({right arrow over (v)}.sub.c[{right arrow over (v)}.sub.o, 0].sup.)({right arrow over (v)}.sub.g[v.sub.0, 0].sup.)({right arrow over (v)}.sub.g[v.sub.0, 0].sup.). Therefore, the difference between {right arrow over (u)} and {right arrow over (d)} can be expressed as

[00010]$\begin{array}{cc}\stackrel{.fwdarw.}{u}-\stackrel{.fwdarw.}{d}=\left[\begin{array}{c}{v}_c^d\\ v_c^q\end{array}\right]\left[\begin{array}{c}.Math.v_g.Math.\cos \left(\right)-v_0\\ .Math.v_g.Math.\sin \left(\right)\end{array}\right]\left[\begin{array}{c}{v}_c^d\\ v_c^q\end{array}\right]-\left[\begin{array}{c}{v}_g\\ .Math.v_g.Math.\end{array}\right].& \left(6a\right)\end{array}$

In Equation (6a), =.sub.g is the angle between v.sub.g and the dq rotating frame. Furthermore, v.sub.g=d.sup.qu.sub. where

[00011]$\begin{array}{cc}{d}^q=.Math.v_g.Math.{}_{t_0}^t\left({\stackrel{}{}}_g\left(\right)-{}_0\right)d,u_{}=.Math.v_g.Math.{}_{t_0}^t\left({\stackrel{}{}}_g\left(\right)-{}_0\right)d.& \left(6b\right)\end{array}$

[0056] Here, the capacitor voltage, {right arrow over (v)}.sub.c, and dq rotating angle, , are controllable, while the grid states, v.sub.g and .sub.g, are not measurable or even controllable. Therefore, Equation (6) can be recovered.

[0057] Some embodiments include a non-transitory computer-readable storage medium storing instructions that, when executed by one or more processors, cause the one or more processors to perform operations for inverter control. In embodiments, the operations include executing a feedback control algorithm that treats power grid connection of one or more inverters, of a multi-source power grid, as a plant comprising resistor-inductor impedance. The operations can further include maintaining, associated with the feedback control algorithm, a two-dimensional operating space parameterized by voltage and frequency control variables associated with the multi-source power grid. In embodiments, the operations include causing, using the feedback control algorithm, transitions between a plurality of operating modes of the one or more inverters by adjusting the voltage and frequency control variables according to grid conditions and power requirements of the multi-source power grid. In embodiments, the operations include ensuring stability of operation of the one or more inverters within the multi-source power grid through satisfaction of sector-boundedness conditions for nonlinear dynamics modeled within the feedback control algorithm.

[0058] Grid disturbance d is unknown; however, the following disturbance model can be adapted for closed-loop analysis, as follows:

[00012]$\begin{array}{cc}\left[\begin{array}{c}{\stackrel{}{d}}^d\\ {\stackrel{}{d}}^q\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& v_0/s\end{array}\right]\left[\begin{array}{c}{v}_g/s\\ {}_g/s\end{array}\right]+\stackrel{.fwdarw.}{\stackrel{}{}}\left(s\right).& \left(7\right)\end{array}$

The above disturbance model assumes nominally constant deviations in voltage and frequency if the power grid, with {right arrow over ()}(s) representing unmodeled disturbances. The model can be updated in Equation (7) if additional information on grid dynamics is available. For example, in the case of a line-to-ground fault, one can include the second-order harmonic model in Equation (7).

[0059] In some embodiments, the closed-loop in FIG. 3 maps the current set-point {right arrow over (.Math.)}.sub.0, and grid disturbance {right arrow over (d)}, as defined in Equation (7), into the grid current {right arrow over (.Math.)}.sub.g and control signal {right arrow over (u)} as

[00013]$\begin{array}{cc}\stackrel{.fwdarw.}{{\stackrel{}{\mathrm{.Math.}}}_g}=G_{L}S\left(K\stackrel{.fwdarw.}{{\widehat{\mathrm{.Math.}}}_0}-\stackrel{.fwdarw.}{\stackrel{}{d}}\right),\stackrel{.fwdarw.}{\stackrel{}{u}}=T\left(G_L^{-1}\stackrel{.fwdarw.}{{\widehat{\mathrm{.Math.}}}_0}+\stackrel{.fwdarw.}{\stackrel{}{d}}\right),\mathrm{where}& \left(8\right)\end{array}$

In Equations (8), S and T represent the sensitivity transfer function and complementary sensitivity transfer function, respectively. associated with the closed-loop system in FIG. 3. Moreover, the S and T transfer functions in Equation (8) are defined in terms of closed-loop plant G.sub.L and feedback controller K shown in FIG. 3 as

[00014]$\begin{array}{cc}S={\left(I_2+K{G}_L\right)}^{-1},T=I_2-S=K{G_L\left(I_2+K{G}_L\right)}^{-1}.& \left(9\right)\end{array}$

It can be clear from Equation (8) that the closed-loop objectives and internal stability may hinge on the sensitivity S and complementary sensitivity T transfer functions in Equation (9). Furthermore, both S and T are shaped by designing a feedback compensator K. In some embodiments, the MIMO feedback compensator K represents a two-by-two matrix of transfer functions with a specific structure.

[0060] In the context of Equation (8) and FIG. 3, the primary challenge in closed-loop analysis and control synthesis arises from the coupled MIMO dynamics of the plant G.sub.L as defined in Equation (4) and feedback controller K. In embodiments, these two MIMO transfer matrices, G.sub.L, and K, form the MIMO sensitivity S per definition in Equation (9). Moreover, the sensitivity S is the primary transfer function that defines the input-output behavior of the closed-loop per Equation (8). In this disclosure, the coupling problem of the MIMO sensitivity S can be mitigated by factoring closed-loop sensitivity S in Equation (9) into diagonal (2-SISO, or single input, single output) sensitivity {tilde over (S)} defined in Equation (10) and a multiplicative coupled perturbation X.sub.c where and X.sub.c are MIMO transfer matrices defined in Equation (10). In this regard, the MIMO sensitivity S=X.sub.c{tilde over (S)}. Subsequently, the bounds for the perturbation X.sub.c can be derived, thus converting the MIMO analysis and synthesis problems into a simpler SISO problem based on the nominal 2-SISO sensitivity {tilde over (S)}, which factorization enables simplifying control. In practical terms, this proposed factorization is like breaking a complex recipe into simpler sub-recipes. The MIMO sensitivity S (which determines how disturbances affect the system output) can be split into: a diagonal part {tilde over (S)} (controlling each axis independently) and a coupling term X.sub.c (representing the interaction between axes). By making the coupling term small, one can design controllers for each axis separately, dramatically simplifying the control problem.

[0061] The disclosed system and method can obtain this factorization by decomposing the controller K into a product of two controller components, {tilde over (K)}K.sub.L, where {tilde over (K)}=diag(K.sup.d, K.sup.q) is a diagonal controller component, and K.sub.L is a stable, proper two-by-two MIMO transfer matrix. The diagonal controller component can be configured to maintain stability and achieve target objectives associated with a decoupled d-axis and a decoupled q-axis under an impact of the modified plant. Also, in the context of FIG. 3, the modified plant can be defined as G.sub.M=K.sub.LG.sub.L and factorized as G.sub.M={tilde over (G)}.sub.M(I.sub.2+E), where

[00015]${\stackrel{}{G}}_M=\mathrm{diag}\left(G_M^d,G_M^q\right)$

includes the diagonal elements of G.sub.M, and

[00016]$E=G_M^{-1}\left(G_M-{\stackrel{}{G}}_M\right)$

represents the coupling terms. The factorization of the MIMO sensitivity transfer function can be summarize as S as follows.

[0062] Proposition 3.2: (a) Factorization of Sensitivity. The MIMO sensitivity S in (9) can be factored into a nominal 2-SISO sensitivity {tilde over (S)}=diag({tilde over (S)}.sup.d, {tilde over (S)}.sup.q) and coupled perturbation X.sub.c as

[00017]$\begin{array}{cc}S={}_c\stackrel{}{S},\mathrm{where}=G_M^{-1}{\stackrel{}{G}}_M,{}_c={\left(I_2+\stackrel{}{S}\left(-I_2\right)\right)}^{-1},\mathrm{and}& \left(10\right)\end{array}$$\stackrel{}{S}={\left(I_2+\stackrel{}{K}{\stackrel{}{G}}_M\right)}^{-1}=\left[\begin{array}{cc}{\left(1+K^d{G}_M^d\right)}^{-1}& 0\\ 0& {\left(1+K^q{G}_M^q\right)}^{-1}\end{array}\right].$

[0063] (b) Coupling Strength. The coupling impact of X.sub.c in Equation (10) between the input and output of the system can be bounded as follows

[00018]$\begin{array}{cc}{.Math.{}_c\left(j\right)-I_2.Math.}_2\frac{\left(\right)}{1-\left(\right)}& \left(11\right)\end{array}$

Here () represents the size of coupling dynamics and captures how small and insignificant the impact is of coupling on system dynamics. In mathematical term () can be defined as ()=S(j).sub.2(j)I.sub.2.sub.2 and it may be desired to keep () small to avoid complications arising from coupled dynamics in the system. The definition of () allows us to effectively design the sensitivity S or to reduce the coupling dynamics as explained below after Equation (14). In embodiments, the robustness is quantified by a coupling strength metric based on deviation from diagonal plant structure sensitivity bounds at fundamental and harmonic frequencies. The robustness can further be quantified by a condition number of modified plant across operating frequency range.

[0064] (c) Closed-Loop Characterization: Effect of the coupling X.sub.c and 2-SISO sensitivity components {tilde over (S)} on closed-loop dynamics in Equation (8) can be expressed as

[00019]$\begin{array}{cc}\stackrel{.fwdarw.}{{\widehat{\mathrm{.Math.}}}_g^{}}={\tilde{G}}_M{}_c\tilde{S}\left(\tilde{K}\stackrel{.fwdarw.}{{\widehat{\mathrm{.Math.}}}_0}-\stackrel{.fwdarw.}{\hat{d}}\right),\stackrel{.fwdarw.}{\tilde{u}}=\tilde{K}{\tilde{G}}_M{}_c\tilde{S}\left(G_M^{-1}\stackrel{.fwdarw.}{{\widehat{\mathrm{.Math.}}}_0}+\stackrel{.fwdarw.}{\hat{d}}\right).& \left(12\right)\end{array}$

Proof 3.2: (a) Using K={tilde over (K)}K.sub.L and G.sub.M=K.sub.LG.sub.L can result in sensitivity expressed as

[00020]$\begin{array}{cc}S={\left(I_2+K{G}_L\right)}^{-1}={\left(I_2+\tilde{K}{K}_L{G}_L\right)}^{-1}={\left(I_2+\tilde{K}{G}_M\right)}^{-1}.& \left(13\right)\end{array}$

[0065] Replacing the modified plant G.sub.M by {tilde over (G)}.sub.M(I.sub.2+E) can result in

[00021]${\left(I_2+\tilde{K}{\stackrel{}{G}}_M\left(I_2+E\right)\right)}^{-1}={\left(I_2+\stackrel{}{S}\tilde{K}{\stackrel{}{G}}_{M}E\right)}^{-1}{\left(I_2+\tilde{K}{\stackrel{}{G}}_M\right)}^{-1}$

The identity {tilde over (S)}{tilde over (K)}{tilde over (G)}.sub.M=I.sub.2{tilde over (S)} in above can be used to obtain

[00022]$\begin{array}{cc}S={\left(I_2+\left(I_2-\stackrel{}{S}\right)E\right)}^{-1}{\left(I_2+\tilde{K}{\stackrel{}{G}}_M\right)}^{-1}.& \left(14\right)\end{array}$

[0066] As final step, E can be replaced by (.sup.1I.sub.2) and factor out .sup.1 to arrive at the factorization in Equation (10). The Neumann series can be employed to achieve

[00023]${}_c-I_2={\left(I_2+\stackrel{}{S}\left(-I_2\right)\right)}^{-1}-I_2=\underset{n=1}{\overset{}{.Math.}}{\left(-\stackrel{}{S}\left(-I_2\right)\right)}^n$

where the right-hand side of the above is bounded by geometric series below

[00024]$\underset{n=1}{\overset{}{.Math.}}{\left(.Math.\left(j\right)-I_2.Math.{.Math.\stackrel{}{S}\left(j\right).Math.}_2\right)}^n=\frac{\left(\right)}{1-\left(\right)}$

[0067] (c) The proof follows from replacing the sensitivity factorization of Equation (10) into Equation (8) and Equation (9). In some embodiments, the operations include factoring multiple input, multiple output (MIMO) sensitivity into diagonal two-single input, single output (2-SISO) sensitivity components and multiplicative coupled perturbation terms. In embodiments, the factorization enables conversion of MIMO analysis into simpler SISO problems than the MIMO analysis. In related embodiments, controlling the transitions between operating modes includes modeling, using direct quadrature formulation, inputs and outputs of the unified algebraic control system using a pair of a single input, single output (SISO) models including sensitivity and a multiplicative perturbation and deriving a coupling between the pair of SISO models. In embodiments, the operations further include factoring a set of primary closed-loop models and a corresponding closed-loop sensitivity model into a set of diagonal parts. The operations can include determining a bounded coupling perturbation between the set of diagonal parts and shaping sensitivity of the set of diagonal parts to minimize the bounded coupling perturbation.

[0068] Based on Equation (11), we can decrease e by (a) reducing the sensitivity {tilde over (S)}, as defined in Equation (10), through a control design for {tilde over (K)}, that represents the diagonal part of feedback controller K, or (b) shaping the coupling term , defined in Equation (10), close to identity. In this work, we assume that the design satisfies .sub.0.1 and that the approximation I.sub.2 is accurate for transient and steady-state analysis. Therefore, the input-output relation in Equation (12) simplifies into

[00025]$\begin{array}{cc}\stackrel{.fwdarw.}{{\stackrel{}{\mathrm{.Math.}}}_g^{}}=\tilde{T}\stackrel{.fwdarw.}{{\stackrel{}{\mathrm{.Math.}}}_0^{}}-{\stackrel{}{G}}_M\stackrel{}{S}\stackrel{.fwdarw.}{\stackrel{}{d}},\stackrel{.fwdarw.}{\stackrel{}{u}}=\stackrel{}{T}\left(G_M^{-1}\stackrel{.fwdarw.}{{\stackrel{}{\mathrm{.Math.}}}_0^{}}+\stackrel{.fwdarw.}{\stackrel{}{d}}\right),\stackrel{.fwdarw.}{{\hat{e}}^{}}=\stackrel{}{S}\left(\stackrel{.fwdarw.}{{\mathrm{.Math.}}_0^{}}+{\stackrel{}{G}}_M\stackrel{.fwdarw.}{\stackrel{}{d}}\right)& \left(15\right)\end{array}$

where

[00026]$\stackrel{.fwdarw.}{{\mathrm{.Math.}}_g^{}}=K_L\stackrel{.fwdarw.}{{\mathrm{.Math.}}_g},\stackrel{.fwdarw.}{{\mathrm{.Math.}}_0^{}}=K_L\stackrel{.fwdarw.}{{\mathrm{.Math.}}_0},{\stackrel{.fwdarw.}{e}}^{}=K_L\stackrel{.fwdarw.}{e}$

and {tilde over (T)} is a 2-SISO complementary sensitivity defined below

[00027]$\begin{array}{cc}\stackrel{}{T}=I_2-\stackrel{}{S}=\stackrel{}{S}\tilde{K}{\stackrel{}{G}}_M=\left[\begin{array}{cc}{\stackrel{}{S}}^d{K}^d{G}_M^d& 0\\ 0& {\stackrel{}{S}}^q{K}^q{G}_M^q\end{array}\right].& \left(16\right)\end{array}$

[0069] The primary transfer matrices in Equation (15) are diagonal as denoted by the tilde () notation. This allows circumvention of the difficulties of MIMO control and develop a simplified but comprehensive system for closed-loop control design and analysis. In practical terms, this factorization technique simplifies the complex multi-input, multi-output (MIMO) control problem into two simpler single-input, single-output (2-SISO) problems. This is analogous to solving two simple equations instead of one complex coupled equation for closed-loop control. The sensitivity transfer matrix S represents how much the system's output deviates from the desired setpoint when disturbances occur. By breaking S into diagonal components ({tilde over (S)}) and coupling terms (.sub.c), we can design controllers independently for voltage (d-axis) and frequency (q-axis) control, then account for their interaction separately.

[0070] In some embodiments, the characterization of MIMO stability in terms of SISO and coupling perturbation stability can be exploited. The SISO factorization of S can be exploited in Proposition 3.2 to propose novel stability criteria for control synthesis.

[0071] Proposition 4.1: In practical terms, stability means the inverter does not oscillate or diverge when subjected to (grid) disturbances, like ensuring a bridge won't collapse under wind loads. The internal stability of the closed-loop controller in FIG. 3 is guaranteed if (a) holds with either (b) or (c), listed as follows.

[0072] (a) SISO Condition: {tilde over (S)}.sup.d, and {tilde over (S)}.sup.q in (10) are stable. This means each axis (voltage and frequency) is individually stable when considered in isolation.

[0073] (b) Decoupling Condition: G.sub.M is triangular or diagonal. Physically, this means the controller K.sub.L has successfully eliminated cross-coupling between voltage and frequency control.

[0074] (c) Magnitude Condition: in (11) satisfies ().sub.<1. This bounds the residual coupling effects to ensure they do not destabilize the system.

[0075] Proof 4.1: The plant G.sub.L and the controller K are stable and minimum phase (we consider stable and minimum phase controllers); hence, the closed-loop in FIG. 3 is internally stable when S is stable. The stability of {tilde over (S)} and .sub.c is sufficient for the stability of S ( is stable if K.sub.L is minimum phase). Condition (a) assumes that {tilde over (S)} is stable; the stability of the coupling term .sub.c is guaranteed through the spectral radius condition ({tilde over (S)}(j)((j)I.sub.2))<1, . The as defined in Equation (11) is an upper bound on the spectral radius; therefore, the magnitude condition in (c) provides a sufficient but conservative guarantee for the stability of .sub.c. The condition in (b) is less conservative than (c) and can directly guarantee that the spectral radius is zero irrespective of the magnitude and shape of {tilde over (S)}. If G.sub.M is triangular, then

[00028]$-I_2=G_M^{-1}{\tilde{G}}_M-I_2$

can be strictly triangular and, consequently, {tilde over (S)}(I.sub.2) can also be strictly triangular. All the eigenvalues of a strictly triangular matrix are zero; hence, irrespective of {tilde over (S)}, a spectral radius can be equal to zero.

[0076] From this theorem, one can guarantee stability of .sub.c by either reducing the magnitude of {tilde over (S)} or shaping the modified plant G.sub.M into a triangular or diagonal matrix. Maintaining two distinct methods can be helpful, since reducing {tilde over (S)} requires a high-gain controller {tilde over (K)} (see Equation (10)), which is not always feasible. For instance, in GFM operation, as we shall see later, the low-frequency gain of the controller is bounded above, limiting the feasible reduction in sensitivity at low frequencies. Moreover, in any case, {tilde over (S)} converges to the identity at high frequencies for a proper feedback controller.

[0077] In some embodiments, the algebraic structure of plant shaping controller can be employed as follows. In scenarios where it is not possible to arbitrarily reduce {tilde over (S)} (e.g., GFM operation), Proposition 4.1(b) imposes a structure K.sub.L such that the modified plant G.sub.M=K.sub.LG.sub.L is triangular or diagonal. To characterize such a structure, one can consider a static row normalized K.sub.L as

[00029]$\begin{array}{cc}{K}_L=\left[\begin{array}{cc}\cos {}_1& -\sin {}_1\\ \sin {}_2& \cos {}_2\end{array}\right]& \left(17\right)\end{array}$

where .sub.1, .sub.2[0, /2]. In the following proposition, the parameters .sub.1 and .sub.2 can be associated with the stability margin of the coupling term .sub.c and the conditioned number of the modified plant G.sub.M=K.sub.LG.sub.L. Subsequently, a transfer function K.sub.L(s) that simultaneously achieves optimal stability margins can be obtained at both low and high frequencies.

[0078] Proposition 4.2: (a) The modified plant G.sub.M(j0) satisfies Proposition 4.1(b) in steady state if .sub.1=.sub.z or .sub.2=.sub.z. Here, .sub.z=arctan X/R where X=L.sub.0 and R are the transmission line reactance and resistance.

[0079] (b) At high frequency (s.fwdarw.) the stability margin of coupling term .sub.c for K.sub.L in (17) converges to

[00030]$\begin{array}{cc}\underset{s.fwdarw.}{\lim }\det \left({}_c\right)=\frac{\cos \left({}_1-{}_2\right)}{\cos {}_1\cos {}_2}.& \left(18\right)\end{array}$

[0080] For .sub.1=0 or .sub.2=0, the stability margin is 1 (optimal) irrespective of the line impedance phase angle .sub.z. Hence, a static .sub.c can achieve an optimal stability margin at low and high frequencies if {.sub.1=z, .sub.2=0} or {.sub.1=0, .sub.2=.sub.z}.

[0081] (c) The condition number of the modified plant G.sub.M=K.sub.LG.sub.L at both low and high frequencies is given as

[00031]$\begin{array}{cc}\underset{.fwdarw.}{\lim }\left(G_M\left(j\right)\right)=\left(G_M\left(j0\right)\right)=\sqrt{\frac{1+.Math.\sin \left(\right).Math.}{1-.Math.\sin \left(\right).Math.}}& \left(19\right)\end{array}$

where =.sub.1.sub.2. A small condition number indicates the robustness of the plant to input uncertainty. A large condition number, however, implies lack of effective control over plant output associated with smallest singular value. We achieve the smallest possible condition number of one for .sub.1=.sub.2. Static K.sub.L cannot satisfy (a) and (b) and achieve the optimal condition number simultaneously, except for resistive lines.

[0082] Proof 4.2: (a) define Z={square root over (X.sup.2+R.sup.2)} and .sub.1=.sub.z then

[00032]$\begin{array}{cc}{G}_M\left(j0\right)=K_L\left(j0\right)G_L\left(j0\right)=\frac{1}{Z}\left[\begin{array}{cc}1& 0\\ \sin \left({}_2-{}_z\right)& 1\end{array}\right]& \left(20\right)\end{array}$

The case for .sub.2=.sub.z follows the same but instead results in an upper triangular matrix.

[0083] (b) The

[00033]$\underset{s.fwdarw.}{\lim }\tilde{S}=I_2$

can result, therefore

[00034]$\underset{s.fwdarw.}{\lim }\det \left({}_c\right)=\underset{s.fwdarw.}{\lim }{\det \left(\right)}^{-1}.$

Moreover, det().sup.1=det(K.sub.L)det(G.sub.L)/det({tilde over (G)}m). Taking into account that det(K.sub.L)=cos(.sub.1.sub.2), one can obtain

[00035]$\begin{array}{cc}\underset{s.fwdarw.}{\lim }{\det \left(\right)}^{-1}=\cos \left({}_1-{}_2\right)\underset{s.fwdarw.}{\lim }\frac{\det \left(G_L\right)}{\det \left({\tilde{G}}_M\right)}=\frac{\cos \left({}_1-{}_2\right)}{\cos \left({}_1\right)\cos \left({}_2\right)}& \left(20a\right)\end{array}$

and the proof is complete.

[0084] (c) For an arbitrary .sub.1, .sub.2[0, /2] can result in

[00036]$\begin{array}{cc}{G}_M\left(j0\right)=\frac{1}{Z}\left[\begin{array}{cc}\cos \left({}_1-{}_z\right)& -\sin \left({}_1-{}_z\right)\\ \sin \left({}_2-{}_z\right)& \cos \left({}_2-{}_z\right)\end{array}\right]& \left(21\right)\end{array}$

and therefore

[00037]$\begin{array}{cc}\stackrel{\_}{}\left(G_M\left(j0\right)\right)=\sqrt{1+.Math.\sin \left({}_1-{}_2\right).Math.},\underset{\_}{}\left(G_M\left(j0\right)\right)=\sqrt{1-.Math.\sin \left({}_1-{}_2\right).Math.}.& \left(21a\right)\end{array}$

[0085] The Proposition 4.2 outlines a design for a static K.sub.L. One can use the same analysis to obtain the transfer matrix K.sub.L(s). One particular choice that satisfies the stability conditions in (a), (b), and the optimal condition number in Equation (19) is

[00038]$\begin{array}{cc}{K}_L\left(s\right)=\frac{{}_m}{s+{}_m}\left[\begin{array}{cc}\frac{L}{Z}s+\cos {}_z& -\sin {}_z\\ \sin {}_z& \frac{L}{Z}s+\cos {}_z\end{array}\right].& \left(22\right)\end{array}$

This K.sub.L result in the diagonal modified plant can be inserted into G.sub.M as

[00039]$\begin{array}{cc}{G}_M=K_L{G}_L=\frac{{}_m/Z}{s+{}_m}{I}_2={\tilde{G}}_M.& \left(23\right)\end{array}$

[0086] Many choices for the transfer matrix K.sub.L(s) satisfy the stability conditions in Proposition 4.2(a) and Proposition 4.2(b). The proposed K.sub.L in Equation (22) is one such a choice that also achieves an optimal condition number. As discussed in more detail later, one can adopt a K.sub.L(s) with a suboptimal condition number that offers more attractive power sharing performance. The plant shaping controller K.sub.L acts like a pre-filter that transforms the inherently coupled dynamics of the power line into a form that is easier to control. When properly designed, K.sub.L makes the d-axis (voltage/reactive power) and q-axis (frequency/active power) behave independently at critical frequencies, eliminating problematic interactions that could cause instability.

[0087] In some embodiments, the feedback controller includes a multiple input, multiple output (MIMO) feedback controller decomposed into a MIMO plant shaping controller component configured to transform line dynamics into a modified plant with reduced coupling and cross-channel interaction between a d-axis for voltage and a q-axis for frequency. The feedback controller can also include a diagonal controller component configured to maintain stability and achieve target objectives associated with a decoupled d-axis and a decoupled q-axis under an impact of the modified plant. The operations may further include implementing a plant shaping controller that satisfies decoupling conditions at steady state through triangular or diagonal structure, stability margin optimization at high frequencies, and/or condition number optimization for robustness to input uncertainty. In some embodiments, the plant shaping controller component is configured to achieve: diagonal modified plant dynamics at steady state; triangular modified plant dynamics at steady state; and/or an optimal condition number for the modified plant across frequency ranges for a highest stability margin.

[0088] In some embodiments, specifying objectives and operating modes based on the closed-loop sensitivity can be employed. In seeking for sensitivity and inverter closed-loop objectives, relevant steady-state control objectives for inverters include

[00040]$\underset{t.fwdarw.}{\lim }\stackrel{.fwdarw.}{v_c}\left(t\right)=0,\underset{t.fwdarw.}{\lim }\left(t\right)=0,\underset{t.fwdarw.}{\lim }{\stackrel{.fwdarw.}{e}}^{}\left(t\right)=0$

[0089] In the context of FIG. 3 and Equation (6), {right arrow over (v)}.sub.c and are inverter capacitor voltage deviation from the nominal value, and inverter frequency deviation from nominal value. These are also components of the closed-loop control effort {right arrow over (u)}, while {right arrow over (e)} represents the modified current feedback error. This observation, together with Equation (15), facilitates formulating inverter control objectives in terms of closed-loop sensitivity {tilde over (S)} and complementary sensitivity {tilde over (T)} as follow

[00041]$\begin{array}{cc}{.Math.\stackrel{.fwdarw.}{u}.Math.}_2{.Math.\tilde{T}.Math.}_2{.Math.G_M^{-1}\stackrel{.fwdarw.}{{\widehat{\mathrm{.Math.}}}_0}+\stackrel{.fwdarw.}{\hat{d}}.Math.}_2,{.Math.\stackrel{.fwdarw.}{{\hat{e}}^{}}.Math.}_2{.Math.\tilde{S}.Math.}_2{.Math.\stackrel{.fwdarw.}{{\widehat{\mathrm{.Math.}}}_0^{}}+{\tilde{G}}_M\stackrel{.fwdarw.}{\hat{d}}.Math.}_2& \left(24\right)\end{array}$

[0090] In some embodiments, it is desirable to minimize {tilde over (T)}.sub.2 and {tilde over (S)}.sub.2 to reduce effect of the exogenous inputs {{right arrow over (.Math.)}.sub.0, {right arrow over (d)}} on {{right arrow over (u)}, {right arrow over (e)}}, thereby improving voltage, frequency, and current regulation. However, the algebraic constraint {tilde over (S)}+{tilde over (T)}=I.sub.2, evident from Equation (16), indicates a fundamental trade-off. This trade-off exists between achieving perfect voltage and frequency regulation (VSI operation) by keeping {tilde over (S)}.sub.2 small and perfect current tracking (GFL operation) by reducing {tilde over (T)}.sub.2.

[0091] FIG. 5 is a continuum representation of a spectrum between GFL and VSI parameterized by {tilde over (S)} according to some embodiments. The GFM operation is based on adjusting the sensitivity {tilde over (S)} to achieve the desired balance between the VSI and GFL operations, as illustrated in FIG. 5. In some embodiments, a harmonic tradeoff can exist between voltage and current. An ideal PR compensator with a resonant frequency of .sub.h in {tilde over (K)}, leads to {tilde over (S)}(j.sub.h)=0 and (j.sub.h)=I.sub.2. This shifts harmonics from line current to capacitor voltage and inverter frequency according to Equation (24). On the other hand, the notch filter in {tilde over (K)}, results in {tilde over (S)}(j.sub.h)=I.sub.2 and {tilde over (T)}(j.sub.h)=0, shifting the harmonics from voltage and frequency to current.

[0092] In some embodiments, line-to-ground fault results in a considerable second harmonic component in grid disturbance {right arrow over (d)}. Overlooked, the disturbance can spread to the line current causing fault trip. A second-harmonic PR compensator ensures {tilde over (S)} (j2120)=0 and {right arrow over (e)}(j2120).sub.2=0 in Equation (24), providing precise control of the line current at 120 Hz.

[0093] In some embodiments, a sensitivity exists between four fundamental operating modes. The inverter's operational mode is determined by how the sensitivity {tilde over (S)} shapes and mitigates the impact of grid disturbance {right arrow over (d)} on the current feedback error {right arrow over (e)} during steady-state operation. To illustrate this, consider the transfer function between the grid disturbance {right arrow over (d)} and {right arrow over (e)} from Equation (15) and expand {right arrow over (d)} according to the nominal disturbance model in Equation (7) to obtain

[00042]$\begin{array}{cc}\left[\begin{array}{c}{\hat{e}}^d\\ {\hat{e}}^q\end{array}\right]={\tilde{G}}_M\tilde{S}\stackrel{.fwdarw.}{\hat{d}}{\tilde{G}}_M\left[\begin{array}{cc}{\tilde{S}}^d& 0\\ 0& {\tilde{S}}^q\end{array}\right]\left[\begin{array}{cc}1/s& 0\\ 0& v_0/s^2\end{array}\right]\left[\begin{array}{c}{v}_g\\ {}_g\end{array}\right]& \left(25\right)\end{array}$

[0094] Based on Equation (25), there are only four meaningful scenarios for steady-state operation. These scenarios depend on whether {tilde over (S)}.sup.d and {tilde over (S)}.sup.q can either completely reject grid disturbances {v.sub.g, .sub.g} in Equation (25) or result in a finite gain between {v.sub.g, .sub.g} and {e.sup.d, e.sup.q} during steady-state operation. In the following proposition, demonstrated herein are that these scenarios correspond to GFL, GFM, ESS and STATCOM operation, and can be characterized by the zeros of {tilde over (S)}.sup.d and {tilde over (S)}.sup.q.

[0095] Proposition 5.1: Inverter operating modes are determined by the number of zeros at the origin in the diagonal components of {tilde over (S)}, denoted by {tilde over (S)}.sup.d and {tilde over (S)}.sup.q in (10), as follows: [0096] (a) GFL: If {tilde over (S)}.sup.d has one zero and {tilde over (S)}.sup.q includes two zeros at the origin, we achieve complete grid disturbance rejection and

[00043]$\begin{array}{cc}\underset{t.fwdarw.}{\lim }{e}^d=0,\underset{t.fwdarw.}{\lim }{e}^q=0& \left(26\right)\end{array}$ [0097] (b) STATCOM: If {tilde over (S)}.sup.d has no zero and {tilde over (S)}.sup.q includes two zeros at the origin, the inverter operates at STATCOM and

[00044]$\begin{array}{cc}\underset{t.fwdarw.}{\lim }{e}^d=v_g/\left({\tilde{G}}_M^{-1}\left(j0\right)+K^d\left(j0\right)\right),\underset{t.fwdarw.}{\lim }{e}^q=0& \left(27\right)\end{array}$ [0098] (c) ESS: If {tilde over (S)}.sup.d and {tilde over (S)}.sup.q each maintain one zero at the origin, the inverter functions as ESS and

[00045]$\begin{array}{cc}\underset{t.fwdarw.}{\lim }{e}^d=0,\underset{t.fwdarw.}{\lim }{e}^q=\underset{s.fwdarw.0}{\lim }{}_g/\left({\mathrm{sK}}^q\left(s\right)\right)& \left(28\right)\end{array}$ [0099] (d) GFM: If {tilde over (S)}.sup.d has no zero at the origin and {tilde over (S)}.sup.q has one zero at the origin, the closed-loop in FIG. 3 leads to an inherent and well-posed drooping characteristic at steady state given below

[00046]$\begin{array}{cc}\left[\begin{array}{c}{v}_g\\ {}_g\end{array}\right]=\left[\begin{array}{cc}{\tilde{G}}_M^{-1}\left(j0\right)+K^d\left(j0\right)& 0\\ 0& {\lim }_{s.fwdarw.0}{\mathrm{sK}}^q\left(s\right)\end{array}\right]\left[\begin{array}{c}{\hat{e}}^d\\ {\hat{e}}^q\end{array}\right]& \left(29\right)\end{array}$

[0100] In some embodiments, the operations further include implementing droop characteristics for power sharing among parallel inverters, of the one or more inverters. Such implementing can include d-axis droop coefficients that determine voltage regulation characteristics. In embodiments, the d-axis droop coefficients are bounded by line impedance characteristics for d-axis voltage droop. Such implementing can include q-axis droop coefficients, associated with grid frequency, that determine frequency regulation characteristics. In embodiments, where the q-axis droop coefficients are bounded by a targeted frequency response bandwidth for q-axis frequency droop.

[0101] Proof 5.1: (a), (b), (c), and (d) are the direct result of application of the final value theorem to Equation (25). FIG. 4A, FIG. 4B, FIG. 4C, FIG. 4D are Bode plots illustrating inverter operating mode based on the shape and number of zeros in {tilde over (S)} at low-frequency in, respectively: (a) grid-forming (GFM); (b) voltage support in synchronous compensator (STATCOM); (c) frequency support in an energy storage system (ESS); and (d) grid-following (GFL). For control design, one can leverage the definition of in Equation (10) to convert the conditions on the number of zeros, as outlined in Proposition 5.1, into an equivalent condition regarding the number of integrators in diagonal components of {tilde over (K)} denoted by K.sup.d and K.sup.q.

[0102] In various embodiments, the control relationships associated with Proposition 5.1 are summarized in Table 1 where the number of integrators (zeros at origin) in the controller determines the inverter's operating mode. The GFL mode may act like a current source, precisely tracking power commands. The GFM mode may act like a voltage source with droop characteristics, supporting grid voltage and frequency stability. The STATCOM mode may provides voltage support through reactive power exchange. The ESS mode may provide frequency support through active power exchange. Each mode represents a different balance between maintaining voltage and frequency stability versus tracking power setpoints.

TABLE-US-00001 TABLE 1 K.sup.q (right): K.sup.d (below): 1 (1/s) 2 (1/s) 0 (1/s) GFM ESS 1 (1/s) STATCOM GFL

[0103] In some embodiments, the operations further include characterizing the plurality of operating modes based on a number of zeros at the origin in sensitivity transfer functions. In various embodiments, such characterization can include that a grid-following mode corresponds to one zero in a d-axis, associated with grid voltage, and two zeros in a q-axis associated with grid frequency; a grid-forming mode corresponds to no zeros in the d-axis and one zero in the q-axis; a STATCOM mode corresponds to no zeros in the d-axis and two zeros in the q-axis; and/or an ESS mode corresponds to one zero in the d-axis and one zero in the q-axis.

[0104] The more accurate term for STATCOM and ESS operation in the Proposition 5.1 is the voltage support and the frequency support, since in steady-state e.sup.d and e.sup.q are linear combinations of e.sup.d and e.sup.q specified by K.sub.L(j0). For example, with K.sub.L in Equation (22), steady-state {right arrow over (e)} can be illustrated as

[00047]$\begin{array}{cc}{\stackrel{.fwdarw.}{e}}^{}=K_L\stackrel{.fwdarw.}{e}.fwdarw.\left[\begin{array}{c}{e}^d\\ e^q\end{array}\right]=\left[\begin{array}{cc}\cos {}_z& -\sin {}_z\\ \sin {}_z& \cos {}_z\end{array}\right]\left[\begin{array}{c}{e}^d\\ e^q\end{array}\right].& \left(30\right)\end{array}$

[0105] In this case, the nature of power transactions for voltage and frequency regulation is a combination of active and reactive power, depending on the characteristics of the line. For inductive lines (.sub.z90), one can observe conventional STATCOM and ESS operation, where voltage regulation is achieved through reactive current transactions and frequency regulation through active current transactions.

[0106] In some embodiments, power sharing exists among parallel GFM inverters. Power sharing between parallel inverters can be pursued for the sustained operation of GFM inverters with varying power ratings and to enforce the DER usage priority. Power sharing can be defined as the change in the active and reactive output power or current from the nominal set point, proportional to changes in the PCC voltage v.sub.g and .sub.g. This proportion for each GFM inverter determines the sharing ratio. Based on this definition and the droop law in Equation (29) the sharing ratio between two arbitrary GFM inverters (i) and (j) is

[00048]$\begin{array}{cc}\frac{{\left({\tilde{G}}_M^{-1}\left(j0\right)+K^d\left(j0\right)\right)}^{\left(i\right)}}{{\left({\tilde{G}}_M^{-1}\left(j0\right)+K^d\left(j0\right)\right)}^{\left(j\right)}}=\frac{e^{d\left(j\right)}}{e^{d\left(i\right)}},\underset{s.fwdarw.0}{\lim }\frac{{K^q\left(s\right)}^{\left(i\right)}}{{K^q\left(s\right)}^{\left(j\right)}}=\frac{e^{q\left(j\right)}}{e^{q\left(i\right)}}& \left(31\right)\end{array}$

[0107] In some embodiments, the d-axis sharing in Equation (31) depends on the controller K.sup.d and the line impedance

[00049]${\tilde{G}}_M^{-1}.$

However, one call reduce the effect of line impedance on power sharing and droop behavior in Equation (29) by increasing the DC gain of the controller

[00050]${\tilde{G}}_M^{-1}\left(j0\right)K^d\left(j0\right).$

This subsumes the concept of virtual impedance.

[0108] As mentioned earlier, the q-axis sharing in Equation (31), unlike the d-axis, depends solely on K.sup.q. This allows for exact sharing of the e.sup.q component between inverters with different line impedances. However, based on the specific choice of K.sub.L, e.sup.q in Equation (31) represents a linear combination of e.sup.d and e.sup.q. For example, K.sub.L(j0) in Equation (22) results in e.sup.q=[sin .sub.z cos .sub.z]{right arrow over (e)}. This particular choice of K.sub.L can lead to exact active power sharing only for inductive lines (.sub.z90). However, once can achieve exact active current sharing by adopting

[00051]$K_L^{*}$

in Equation (32), which maps e.sup.q to e.sup.d at steady-state, regardless of line impedance.

[00052]$\begin{array}{cc}{K}_L^{*}\left(s\right)=\frac{{}_m}{s+{}_m}\left[\begin{array}{cc}\frac{L}{Z}s+\cos {}_z& -\sin {}_z\\ \frac{}{{}^2+{}_0^2}s+1& \frac{{}_0}{{}^2+{}_0^2}s\end{array}\right]& \left(32\right)\end{array}$

[0109] The proposed

[00053]$K_L^{*}$

satisfies stability criteria outlined in parts (a) and (b) of the Proposition 4.2, resulting in the following modified plant G.sub.M* and its diagonal counterpart

[00054]${\tilde{G}}_M^{*}$

below

[00055]$\begin{array}{cc}{G}_M^{*}=\frac{{}_m/Z}{s+{}_m}\left[\begin{array}{cc}1& 0\\ \cos {}_z& \sin {}_z\end{array}\right],{\tilde{G}}_M^{*}=\frac{{}_m/Z}{s+{}_m}\left[\begin{array}{cc}1& 0\\ 0& \sin {}_z\end{array}\right].& \left(33\right)\end{array}$

[0110] However,

[00056]$G_M^{*}\left(j0\right)$

in Equation (33) has a suboptimal condition number of

[00057]$\left(G_M^{*}\left(j0\right)\right)=\left(1+\cos \left({}_z\right)\right)/\sin \left({}_z\right)\mathrm{and}\mathrm{for}{}_{z}11,$

(dominantly resistive line), the condition number becomes exceeds 10. This suggests that

[00058]$K_L^{*}$

trades the robustness margin to achieve exact active power-sharing.

[0111] In some embodiments, synchronization and frequency transient responses can be explained as follows. Existing GFL and STATCOM use the PLL mechanism to synchronize their rotating frame with an established grid voltage. However, GFM and ESS synchronize through power transactions based on frequency droop. Therefore, changing between the GFL/STATCOM and GFM/ESS modes involves changing the synchronization method. In this section, we exploit the parallel structure of the q-axis controller, that is

[00059]$K^q=K_1^q+K_2^q$

in FIG. 3 to develop a mode-independent synchronization scheme and generalize the concept of inertia and rate-of-change of frequency (RoCoF) as a closed-loop transient response. In the context of the closed-loop signals in FIG. 3 the two primary synchronization conditions are

[00060]$\begin{array}{cc}\left(34\right)& \\ \underset{t.fwdarw.}{\lim }{v}_c^q=0,& \left(a\right)\end{array}$$\begin{array}{cc}\underset{t.fwdarw.}{\lim }{v}_0\left({\stackrel{.}{}}_g-\stackrel{.}{}\right)\underset{t.fwdarw.}{\lim }{\stackrel{.}{d}}^q-u_{}=0& \left(b\right)\end{array}$

where {dot over ()}.sub.g is the time derivative of grid voltage angle (grid frequency), and {dot over ()} is the time derivative of inverter synchronous frame angle (inverter frequency). The approximation in (b) is based on definition of dq and u.sub. in Proposition 3.1.

[0112] Condition (a) aligns the capacitor voltage phasor {right arrow over (v)}.sub.c with the dq frame while (b) ensures that the rotating frame achieves frequency lock with the grid frequency {dot over ()}.sub.g. Both conditions are expressed in terms of the closed-loop variables

[00061]$v_c^q$

and u.sub., along with the grid disturbance dq (see FIG. 3). This allows conversion of synchronization conditions in Equation (34) into constraints for the feedback control design. To clarify, consider the transfer function between the exogenous inputs, and

[00062]$v_c^q$

and u.sub. as follows:

[00063]$\begin{array}{cc}{\hat{v}}_c^q=T_v\left(\hat{w}+{\hat{d}}^q\right),{\hat{u}}_{}=T_{}\left(\hat{w}+{\hat{d}}^q\right),& \left(35\right)\end{array}$

where

[00064]$w=\left[\begin{array}{cc}0& 1\end{array}\right]G_M^{-1}\stackrel{.fwdarw.}{{\widehat{\mathrm{.Math.}}}_l^{}}$

and the transfer functions T.sub.v and T.sub. are

[00065]$\begin{array}{cc}{T}_v=\frac{G_M^q{K}_1^q}{1+G_M^q\left(K_1^q+K_2^q\right)},T_{}=\frac{G_M^q{K}_2^q}{1+G_M^q\left(K_1^q+K_2^q\right)}.& \left(36\right)\end{array}$

[0113] Using final value theorem and Equation (35), Equation (34) can be converted into

[00066]$\begin{array}{cc}\underset{s.fwdarw.0}{\lim }{\mathrm{sT}}_v\left(\hat{w}+{\hat{d}}^q\right)=0,\underset{s.fwdarw.0}{\lim }{s}^2\left(\left(1-T_{}\right){\hat{d}}^q-T_{}\hat{w}\right)=0& \left(37\right)\end{array}$

[0114] In some embodiments, the operations include implementing mode-independent synchronization by maintaining synchronization conditions across the plurality of operating modes by ensuring capacitor voltage alignment with a direct-quadrature (dq) reference frame and achieving a frequency lock between the inverter and a grid frequency.

[0115] FIG. 6 is a Bode plot illustrating operations of desired q-axis closed-loops T.sub.v, T.sub., {tilde over (T)}.sup.q, and corresponding open-loops

[00067]$K_1^q{\tilde{G}}_M,K_2^q{\tilde{G}}_M$

and K.sup.q{tilde over (G)}.sub.M according to some embodiments. The equation above allows us to formulate the synchronization in terms of the conditions on the controllers

[00068]$K_1^q\mathrm{Kq}\mathrm{and}{K}_2^q.$

[0116] Proposition 5.2: We achieve synchronization, regardless of the operating mode, if

[00069]$\left(K_1^q/K_2^q\right)$

includes at least two zeros at the origin and

[00070]$K_2^q$

has at least one pole at the origin.

[0117] Proof 5.2: Adopting the nominal ramp model in Equation (7) for d.sup.q and considering

[00071]$T_v=T_{}{K}_1^q/K_2^q,$

Equation (37) is rewritten as follows

[00072]$\underset{s.fwdarw.0}{\lim }s\frac{K_1^q}{K_2^q}{T}_{}\left(\hat{w}+v_0\frac{{}_g}{s^2}\right)=0,\underset{s.fwdarw.0}{\lim }{v}_0\left(1-T_{}\right){}_g-s^2{T}_{}\hat{w}=0.$

Based on the conditions in the proposition lim.sub.s.fwdarw.0T.sub.=1, lims.fwdarw.0 T=1, and

[00073]${\lim }_{s.fwdarw.0}{K}_1^q/\left(s^2{K}_2^q\right)=const,$

completing the proof. In some embodiment, the synchronization is achieved by configuring a first q-axis controller with at least two zeros at the origin relative to a second q-axis controller and the second q-axis controller with at least one pole at the origin.

[0118] 1) Transient response of Inverter Frequency: T.sub. in Equation (36) represents the closed-loop transfer function between the inverter's frequency deviation , and the grid frequency deviation .sub.g. Therefore, the characteristics of T.sub., such as the bandwidth, generalize the concepts of inertia and RoCoF, while the peak of T.sub., denoted as M.sub.T=T.sub..sub., is directly proportional to the overshoot (frequency nadir) and the transient oscillation of . Therefore, it is desired to shape T.sub. as a unity gain low-pass filter with M.sub.T<3 dB, and low bandwidth. However, there are two sources of difficulties. First, the transient frequency response, the current and the voltage regulation objectives on the q axis are interdependent and conflicting. Second, shaping T.sub. and T.sub.v in (36) to achieve the desired closed-loop objectives requires the concurrent design of both

[00074]$K_1^q\mathrm{and}{K}_2^q$

[0119] We establish the design criteria for

[00075]$K_1^q\mathrm{and}{K}_2^q$

considering the fundamental algebraic constraints outlined below

[00076]$\begin{array}{cc}{T}_{}+T_v={\stackrel{}{T}}^q,{\stackrel{}{S}}^q+{\stackrel{}{T}}^q={\stackrel{}{S}}^q+\left(T_{}+T_v^q\right)=1& \left(38\right)\end{array}$

where {tilde over (T)}.sup.q is the closed-loop variable between the current setpoint

[00077]$i_0^q\mathrm{and}{i}_g^{q}g.$

We aim to keep e{tilde over (T)}.sup.q close to one and {tilde over (S)}.sup.q small over a broad frequency range [0, .sup.q] to achieve effective disturbance rejection in Equation (15), negligible tracking error in Equation (24), dynamic decoupling in Equation (11), and MIMO stability (Proposition 4.1(c)). Furthermore, T.sub. in Equation (36) should be designed as a low-pass filter with a bandwidth .sub.J<<.sup.q to achieve an inertial response, low RoCoF, and to attenuate the effect of high-frequency grid disturbances on . Based on Equation (38), this automatically constrains T.sub.v to a unity gain band-pass filter between .sub.J and .sub.q as shown in FIG. 6. In the following, we sketch the outline of the open-loops M that result in the desired T.sub., T.sub.v and {tilde over (T)}.sup.q Subsequently, in Section VI-A, we provide an example control design for

[00078]$K_1^q\mathrm{and}{K}_2^q.$

[0120] In some embodiments, the operations further include implementing virtual inertia control by shaping a frequency response transfer function as a low-pass filter with adjustable bandwidth and damping characteristics to control: a rate of change of frequency (RoCoF) between the inverter and the grid-tied line; and frequency nadir following disturbances bypassing a reliance on mimicking synchronous generator dynamics.

[00079]$K_1^q{\stackrel{}{G}}_M:$

should be designed as a bandpass filter with a gain cross-over frequency near .sub.q and a phase margin of 60. The pass band should satisfy .sub.J[.sub.1, .sub.2], as shown in FIG. 6, where .sub.J is the bandwidth of T.sub. and adjusts the desired inertial response. The ratio .sub.J/.sub.1 is directly proportional to M.sub.T and damps and reduces the frequency nadir and oscillation. Finally, the filter should achieve slope of 20 dB/dec at low frequency to satisfy synchronization condition in Proposition 5.2.

[00080]$K_2^q{\stackrel{}{G}}_M:$

should be shaped close to an integrator (e.g., .sub./s) that intersects

[00081]$K_1^q{\stackrel{}{G}}_M$

at .sub.J in the pass band region, as shown in FIG. 6. To attenuate the effect of high-frequency disturbances on inverter frequency, we augment the open-loop with a low-pass filter with cut-off frequency .sub.f above .sub.J.

[0121] Hereinafter is disclosed a feedback control design for {tilde over (K)} that meets the stability and performance criteria proposed in this disclosure. Subsequently, we implement the proposed control design using inverter closed-loop dynamics. For the control design, assume the nominal line impedance of Z.sub.z, and adopt K.sub.L in Equation (22) and the corresponding {tilde over (G)}.sub.M in (23) to satisfy the coupling stability in Proposition 4.2. Subsequently, for

[00082]$\tilde{K}=\mathrm{diag}\left(K^d,K_1^q+K_2^q\right)$

we employ the following set of controllers

[00083]$\begin{array}{cc}\begin{array}{c}{K}^d=\frac{s+{}_m}{{}_m/Z}\left(\frac{s+{}_v}{s+2\sqrt{2}\left(Z\right){}_v}\right){\left(\frac{\sqrt{2}{}_d}{s+{}_d}\right)}^3\left(\frac{s+{}_d}{s+{}_d}\right)\\ K_1^q=\frac{s+{}_m}{{}_m/Z}\left(\frac{\sqrt{{}_q^2+{}_2^2}s}{\left(s+{}_1\right)\left(s+{}_2\right)}\right){\left(\frac{\sqrt{2}{}_q}{s+{}_q}\right)}^2\left(\frac{s+{}_q}{s+{}_q}\right)\\ K_2^q=\frac{s+{}_m}{{}_m/Z}\left(\frac{{}_{}}{s}\right)\left(\frac{s+{}_{}/{}_{}}{s+{}_{}Z}\right)\left(\frac{{}_f}{s+{}_f}\right)\end{array}& \left(39\right)\end{array}$

[0122] We justify the proposed control parameters and design by considering their dynamic and steady-state characteristics as follows.

[0123] 1) d-axis: Droop characteristics: Based on Equation (29), the steady-state voltage droop coefficient for K.sup.d in (39) is

[00084]$\begin{array}{cc}{\stackrel{}{G}}_M^{-1}+K^d\left(j0\right)=Z+\frac{{}_v}{{}_{}},\max \left\{\frac{{}_v}{aZ},\frac{4\left(aZ\right)}{\sqrt{2}}{}_v\right\}<<{}_d.& \left(40\right)\end{array}$

The voltage droop depends on the ratio .sub.v/.sub.v. Higher values of .sub.v and .sub.v correspond to faster zero and pole in Equation (39) and quicker droop responses. We recommend limiting .sub.v and .sub.v by the d-axis open-loop bandwidth as indicated in (40).

[0124] Dynamic characteristics: Given that inequality in Equation (40) holds, K.sup.d in (39) leads to an open-loop

[00085]$K^d{G}_M^d$

with a gain crossover frequency of close to .sub.d and phase margin

[00086]$\begin{array}{cc}PM45+\mathrm{arc}\sin \left(1-a\right)/\left(1+a\right),0a1& \left(41\right)\end{array}$

[0125] 2) q-axis: Droop characteristics: Based on (29), the steady-state frequency droop coefficient for K.sup.q in Equation (39) is

[00087]$\begin{array}{cc}\underset{s.fwdarw.0}{\lim }{\mathrm{sK}}^q=\underset{s.fwdarw.0}{\lim }s\left(K_1^q+K_2^q\right)=\frac{{}_{}}{{}_{}},\max \left\{\frac{{}_{}}{{}_{}},{}_{}\right\}<<{}_J& \left(42\right)\end{array}$

We limit .sub. and .sub. by .sub.J in (42), where .sub.J represents the bandwidth for T.sub. to achieve desired inertia.

[0126] Dynamic characteristics

[00088]$K_1^q\mathrm{and}{K}_2^q$

in Equation (39) satisfy the synchronization condition in Proposition 5.2. Furthermore, for .sub.1<.sub.2<<.sup.q the shape of

[00089]$K_1^q{\tilde{G}}_M$

traces the desired shape shown in FIG. 6, with a gain cross-over frequency of .sup.q and a phase margin close to PMarcsin(1a)/(1+a). The proposed

[00090]$K_2^q$

in Equation (39), includes an integrator .sub./s, lowpass filter with a cut-off frequency of .sub.f(.sub.J<.sub.f) to attenuate the disturbance in the inverter frequency, and a compensator to shape the drooping characteristics. The resulting

[00091]$K_2^q{\tilde{G}}_M$

intersects the pass band region of

[00092]$K_1^q{\tilde{G}}_M$

at .sub.J[.sub.1, .sub.2] provided that .sub. satisfies the constraint

[00093]$\begin{array}{cc}\log \left({}_J\right)-\log \left({}_2\right)\log \left({}_{}\right)-\log \left(a{}_q\right),{}_{}\frac{{}_q}{a{}_2}{}_J.& \left(43\right)\end{array}$

This allows us to calculate .sub. for the desired inertial bandwidth .sub.J. Finally, by adjusting .sub.1, we can fine-tune the ratio of .sub.J/.sub.1 for the desired damping of the frequency oscillation.

[0127] 3) PR compensator for enhanced stability: Proposition 9.1 imposes an upper-bound on the 2-SISO sensitivity {tilde over (S)} to ensure stability in the presence of the nonlinear term (, ) in line dynamics.

[0128] FIG. 7 is a Bode plot illustrating smaller values of , which impose stricter upper-bounds on sensitivity around the fundamental frequency .sub. according to at least one embodiment. This upper bound is explicitly dependent on G.sub.L(j).sup.1, where the shape of G.sub.L(j) is uniquely determined by the line characteristics, denoted by =R/L. As shown in FIG. 7, smaller values of result in a significant notch in G.sub.L(j).sup.1 at the fundamental frequency .sub.0. Consequently, this requires the 2-SISO sensitivity transfer function {tilde over (S)} to be small at 0. In embodiments, the sensitivity transfer function {tilde over (S)}(j) denotes the current error signal (e.g., tracking error) relative to the desired current setpoint at frequency , which can be understood as an algebraic difference between the setpoint and the measured output relative to setpoint at given frequency . Thus, {tilde over (S)}(j)=({circumflex over (r)}(j)(j))/{circumflex over (r)}(j)), where {circumflex over (r)}(j) is the setpoint or reference signal value at frequency and (j) is the measured process variable, e.g., actual frequency actual voltage, actual power output component at frequency . Based on the definition of {tilde over (S)} in Equation (10), we can reduce {tilde over (S)} at 0 by cascading a proportional resonant compensator (PR) in series K.sup.d and

[00094]$K_1^q$

controllers in Equation (39). An example of a PR compensator is given below

[00095]$\begin{array}{cc}{K}_{\mathrm{PR}}=1+k_{{}_0}\frac{2{}_{0}s}{s^2+2{}_{0}s+{}_0^2},=\frac{{}_{\max }}{{}_0}& \left(44\right)\end{array}$

where k.sub..sub.0 sets the PR gain and .sub.max represents the maximum expected frequency deviation from nominal value.

[0129] We simplified closed-loop analysis and design by neglecting the dependence of K.sup.d and

[00096]$K_1^q$

on the inverter closed-loop dynamics. In the following discussion related to FIGS. 8A-8B, discussed is how to use inverter dynamics to implement K.sup.d and

[00097]$K_1^q.$

[0130] In some embodiments, an inverter closed-loop can function as a feedback compensator as follows. We adopt the cascaded closed-loop structure in FIG. 8A with inner current and outer voltage loops. Here, G.sub.i=1/(L.sub.is+R.sub.i) and G.sub.v=1/(C.sub.is) represent the dynamics of the inductor and capacitor in Equation (5). To form the inner closed loop, we use feedback linearization to decouple the inductor dynamics in Equation (5) by defining the modulation signal as

[00098]$\begin{array}{cc}\stackrel{.fwdarw.}{m}=\frac{2}{v_{\mathrm{dc}}}\left({\stackrel{.fwdarw.}{u}}_{\mathrm{.Math.}}+{\stackrel{.fwdarw.}{v}}_c+L_i{\stackrel{.}{}\left[-i_L^q,i_L^d\right]}^{}\right)& \left(45\right)\end{array}$

where {right arrow over (u.sub..Math.)} denotes the output of current controller K.sub.i in FIG. 8A. Subsequently, we use the proportional-integral (PI) controller K.sub.i=.sub.c(L.sub.is+R.sub.i)/s to shape the inner closed-loop Ti in FIG. 8A into a unity gain low-pass filter with cut-off frequency .sub.c below

[00099]$\begin{array}{cc}{T}_i=\frac{G_i{K}_i}{1+G_i{K}_i}=\frac{{}_c}{s+{}_c},\mathrm{and}{S}_i=1-T_i=\frac{s}{s+{}_c}& \left(46\right)\end{array}$

[0131] The operations can further include implementing, as part of the unified algebraic control system, a cascaded closed-loop structure with an inner current loop and an outer voltage loop, as illustrated in FIGS. 8A-8B. The operations can include using feedback linearization to decouple inductor dynamics of an inductor and a capacitor being modeled within the inner current loop and employing a phase interpolator controller to shape the inner current loop into a unity gain low-pass filter.

[0132] Specifically, FIG. 8A and FIG. 8B are circuit diagrams illustrating, respectively, (a) nested structure with voltage and current inputs; and (b) inverter closed-loop as part of feedback compensator. The outer voltage loop is formed by setting the reference,

[00100]$\stackrel{.fwdarw.}{{\mathrm{.Math.}}_L^{*}},$

for the inner closed-loop as follows

[00101]$\begin{array}{cc}\stackrel{.fwdarw.}{{\mathrm{.Math.}}_L^{*}}=\stackrel{.fwdarw.}{{\mathrm{.Math.}}_r}+\stackrel{.fwdarw.}{u_v}+\stackrel{.fwdarw.}{{\mathrm{.Math.}}_g}+C_i{\stackrel{.}{}\left[-v_c^q,v_c^d\right]}^{}& \left(47\right)\end{array}$

where {right arrow over (u.sub.v)} is the output of the voltage compensator K.sub.v and {right arrow over (.Math..sub.r)} is the input to the cascaded closed-loop as shown in FIG. 8A. The feed-forward term

[00102]$\stackrel{.fwdarw.}{{\mathrm{.Math.}}_g}+C_i{\stackrel{.}{}\left[-v_c^q,v_c^d\right]}^{}$

decouples the capacitor dynamics and counteracts the effect of grid current in (5). The proposed cascaded closed-loop in FIG. 8A leads to the following relation between the inputs {{right arrow over (v.sub.0)}, {right arrow over (.Math..sub.r)}, {right arrow over (.Math..sub.g)}}{right arrow over ({circumflex over (v)}.sub.c )}

[00103]$\begin{array}{cc}\stackrel{.fwdarw.}{{\stackrel{}{v}}_c}=G_v{S}_v\left(T_i\stackrel{.fwdarw.}{{\hat{l}}_r}+T_i{K}_v\stackrel{.fwdarw.}{{\stackrel{}{v}}_0}-S_i\left(\stackrel{.fwdarw.}{{\stackrel{}{\mathrm{.Math.}}}_g}+C_i{\stackrel{}{}\left[-v_c^q,v_c^d\right]}^{}\right)\right)& \left(48\right)\end{array}$

where {S.sub.i, T.sub.i} are given by Equation (46), and S.sub.v is the sensitivity transfer function for the voltage closed-loop defined below

[00104]$\begin{array}{cc}{S}_v=\left[\begin{array}{cc}{S}_v^d& 0\\ 0& S_v^q\end{array}\right]\left[\begin{array}{cc}{\left(1+G_v{T}_i{K}_v^d\right)}^{-1}& 0\\ 0& {\left(1+G_v{T}_i{K}_v^q\right)}^{-1}\end{array}\right].& \left(49\right)\end{array}$

[0133] For a large value of .sub.c, S.sub.i in (48) remains small over a broad frequency range and can be assumed to be negligible. Therefore, considering {right arrow over (v.sub.c)}={right arrow over (v.sub.c)}{right arrow over (v.sub.0)}, we can rewrite (48) as

[00105]$\begin{array}{cc}\stackrel{.fwdarw.}{{\stackrel{}{v}}_c}=K_{\mathrm{inv}}{\stackrel{\stackrel{.fwdarw.}{}}{\mathrm{.Math.}}}_r-S_v\stackrel{.fwdarw.}{{\stackrel{}{v}}_0}\mathrm{where}{K}_{inv}=G_v{S}_v{T}_i& \left(50\right)\end{array}$

[0134] In some embodiments, K.sub.inv captures the inverter closed-loop dynamics as a compensator between {right arrow over (.Math..sub.r)} and {right arrow over (v.sub.c)} as shown in FIG. 8B. K.sub.inv can represent any transfer function with a relative degree of at least two and the following zpk (zeros (z), poles (p) and gain (k)) form

[00106]$\begin{array}{cc}{K}_{inv}=\frac{N\left(s\right)}{D\left(s\right)}=\frac{{}_c}{C_i}\frac{\left(s+z_n\right).Math.\left(s+z_0\right)}{\left(s+p_m\right).Math.\left(s+p_0\right)},n\left(m-2\right)& \left(51\right)\end{array}$

if voltage compensator K.sub.v=N.sub.v(s)/D.sub.v(s) is designed as

[00107]$\begin{array}{cc}{N}_v\left(s\right)=D\left(s\right)-C_{i}s\left(s+{}_c\right)N\left(s\right)/{}_c,D_v\left(s\right)=N\left(s\right)& \left(52\right)\end{array}$

and the inner closed-loop controller K.sub.i is set to

[00108]$\begin{array}{cc}{K}_i={}_c\frac{L_{i}s+R_i}{s}\mathrm{where}& \left(53\right)\end{array}$${}_c=\underset{i=0}{\overset{m}{.Math.}}{p}_i-\underset{i=0}{\overset{n}{.Math.}}{z}_i$

where p.sub.i and z.sub.i are poles and zeros of transfer function K.sub.inv in Equation (51). We can cascade K.sub.inv with a series compensator Kc to form the feedback compensator between {right arrow over (e)} and {right arrow over (v.sub.c)} shown in FIG. 8B. In this case, comparing FIG. 3 and FIG. 8B shows

[00109]$\begin{array}{cc}{K}_c{K}_{inv}=\left[\begin{array}{cc}{K}_c^d{K}_{inv}^d& 0\\ 0& K_c^q{K}_{inv}^q\end{array}\right]=\left[\begin{array}{cc}{K}^d& 0\\ 0& K_1^q\end{array}\right]& \left(54\right)\end{array}$

[0135] Equation (53) indicates that poles and zeros of the controllers K.sup.d and

[00110]$K_1^q$

should be divided between K.sub.c and K.sub.inv. We suggest that the inverter dynamics K.sub.inv only includes the smallest zero and three of the largest poles of K.sup.d and

[00111]$K_1^q$

and K.sub.c include rest of poles and zeros. For example, K.sup.d and

[00112]$K_1^q$

in (39) can be divided into K.sub.inv and K.sub.c as follows

[00113]$\begin{array}{cc}{K}_{inv}^d=\frac{{}_c^d}{C_i}\frac{s+{}_v}{{\left(s+{}_d\right)}^3},K_{inv}^q=\frac{{}_c^q}{C_i}\frac{s}{\left(s+{}_2\right){\left(s+{}_q\right)}^2}& \left(55\right)\end{array}$$\begin{array}{cc}{K}_c^d=\frac{C_i}{{}_c^d}\left(\frac{2\sqrt{2}{}_d^3}{{}_m/Z}\right)\left(\frac{s+{}_m}{s+2\sqrt{2}\left(aZ\right){}_v}\right)\left(\frac{s+a{}_d}{as+{}_d}\right)& \left(56\right)\end{array}$$\begin{array}{cc}{K}_c^q=\frac{C_i}{{}_c^q}\left(\frac{2{}_q^2\sqrt{{}_q^2+{}_2^2}}{{}_m/Z}\right)\left(\frac{s+{}_m}{s+{}_1}\right)\left(\frac{s+a{}_q}{as+{}_q}\right)& \left(57\right)\end{array}$

where

[00114]${}_c^d=3{}_d-{}_v\mathrm{and}{}_c^q=2{}_q-{}_2.$

Finally, we implement K.sub.inv in (55) by designing K.sub.v and K.sub.i as

[00115]$\begin{array}{cc}{K}_v^d=C_i\frac{\left(3{}_d^2+{}_v{}_c^d\right)s+{}_d^3}{{}_c^d\left(s+{}_v\right)},K_i^d={}_c^d\frac{L_{i}s+R_i}{s}& \left(58\right)\end{array}$$\begin{array}{cc}{K}_v^q=C_i\frac{\left({}_q^2+2{}_2\right)s+{}_2{}_q^2}{{}_c^{q}s},K_i^q={}_c^q\frac{L_{i}s+R_i}{s}& \left(59\right)\end{array}$